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C-axis oriented barium ferrite thin/thick films for microwave applications

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C- Axis Oriented Barium Ferrite Thin/Thick
Films for Microwave Applications
A Dissertation
Presented in Partial Fulfillment o f the Requirements for the
Degree o f Doctor o f Philosophy
with a
M ajor in Physics
in the
College o f Graduate Studies
University o f Idaho
by
Alaaedeen R. Abuzir
December 2006
Major Professor: Wei Jiang Yeh, Ph.D.
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UMI N um ber: 3250624
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AUTHORIZATION TO SUBMIT DISSERTATION
This dissertation of Alaaedeen R. Abuzir, submitted for the degree of Doctor of
Philosophy with a major in Physics and titled “C-Axis Oriented Barium Ferrite
Thin/Thick Films for Microwave Applications,” has been reviewed in final form.
Permission, as indicated by the signatures and dates given below, is now granted to
submit final copies to the College o f Graduate Studies for approval.
Major Professor
Date
L -t-J
/rJ/ f /
c &
Wei Jiang Yeh
Committee
Members
Date
ouni
Date
David N. Mcllroy
..
Date
Francesca Sammarruca
Department
Administrator
Z
Date
Wei Jiang Yeh
Discipline’s
College Dean
Date
udith Totman Parrish
Final Approval and Acceptance by the College of Graduate Studies
^
Date
^
^ ^
grit von Braun
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ABSTRACT
Because of their good chemical stability and high uniaxial anisotropies, highly caxis oriented hexagonal M-type ferrites (or BaM) have provoked much interest in recent
years for their applications in the microwave devices. In current technology, microwave
ferrite devices use bulk BaM under an external magnetic field bias to control the
microwave propagation in the circulator. Our goal was set to fabricate a self-biased
microwave circulator, which does not require an external magnetic field; therefore, the
self-biased circulator will be much lighter and smaller in size. To develop the future
generation o f microwave devices that can be fully integrated into a chip, and functional in
the GHz range, excellent c-axis orientation with high coercivity, good squareness and
thickness o f at least 100 micrometers or more are required so that BaM films can be self­
biased.
We have used RF sputtering to grow BaM thin and thick films on different
substrates. We have grown BaM thin films using external and in-situ annealing. The caxis orientation o f the films was not o f high quality. With the in-situ method, Ms value
has increased up to 82% o f the bulk value, however the c-axis orientation did not improve
much. Excellent c-axis orientation has been obtained by developing the alternating
temperature multilayered method. Using this method, we grew the antiferromagnetic
phase (a-FeiCb) and used it as a pinning layer to grow BaM thin films on AI2O3 (0001)
without any seed layer. The c-axis and magnetic parameters are consistent with our
mathematical micro-magnetic model.
Liquid Phase Epitaxy (LPE) method has proven to be the most effective way to grow
thick BaM films; these thick films have excellent c-axis orientation and single crystal
structure with about 10 Oe coercivites, however, single crystal BaM films can not be self­
biased. We have developed the vacuum LPE and the reflow method to grow up to 0.5
mm poly crystalline thick films. Our new method uses no seed layer which greatly
simplifies the fabrication process. The effect o f BaM thin film seed layer on the growth
and magnetic properties of the thick films has been briefly studied.
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Acknowledgements
I would like to express my deepest gratitude and appreciation to my major advisor
Prof. Wei Jiang Yeh for his guidance, support, continued supervision throughout this
study, and for being more than an advisor to me. I thank my committee members Prof.
David N. Mcllroy, Prof. Jeffrey L. Young, and Prof. Francesca Sammarruca for their
valuable suggestions, comments and feedback. I would like to thank the U.S. Office o f
Naval Research (ONR) for their funding and financial support. I would like to express
my gratitude to Dr. Yanko Kranov for his collaborative work and help during the second
part of this dissertation and for his help in the MOKE system. I thank Prof. Richard B.
Wells and Dr. Feng Xie for their big help in the micro-magnetic simulation part o f this
dissertation. I thank Mr. Ryan Adams for his suggestions.
I would also like to thank Dr. Daqing Zhang for his help; also I thank Dr. Lidong
Wang for his help in the AFM measurements. I thank Prof. Y. Qiang for his suggestions.
I would like to thank Prof. Y. K. Hong, S. H. Gee and Jeevan Jalli for their help in the
VSM measurements. I thank Dr. Tom Williams and Mr. J. Franklin Bailey for their help
in the X-ray Diffraction and Scanning Electron Microscopy. My sincere thanks go to
Luanne Semler and John Failla for their expertise and assistant. My colleagues Kun
Yang, Shilpa Chava, Ahmad Abu Abdu, Dr. Abdullah Alkhateeb, and Dr. Ehab Marji
deserve special thank for creating a friendly environment, and for being good friends
during the years in the University o f Idaho.
Finally, my sincere and deepest gratitude goes to my parents for their continuous
support, love, encouragement, and prayers. I will be always grateful to what you have
done to me.
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V
Dedication
To My Parents, I can’t Pay You Back
.................To My Country, W e Will Be Back
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vi
Table of Contents
Authorization to Submit Dissertation..................................................................................... ...ii
Abstract..........................................................................................................................................iii
Acknowledgements...........................
iv
Dedication...................................................................................................................................... v
Table o f Contents......................................................................................................................... vi
List of Figures............................................................................................................................ viii
CHAPTER 1 .....................................................................................
1
Introduction.....................................................................................................................................1
1.1 Magnetism in M aterials.....................................................................................................2
1.2 D iam agnetism .....................................................................................................................3
1.3 Paramagnetism................................................................................................................
3
1.4 Ferrom agnetism ..................................................................................................................5
1.5 Antiferromagnetism
..................................................................................................... . 6
1.6 Ferrimagnetism ...................
7
1.6.1 Spinel Ferrites..........................
8
1.6.2 Garnet F errites...................................................
9
1.6.3 Hexagonal Ferrites....................................................................................................10
1.7 The Y-Junction Circulator.............................................................................................. 15
CHAPTER 2 .....................................
19
Experimental Procedure..............................................................................................................19
2.1 Making o f BaM T arget................
2.2 The RF Magnetron Sputtering
19
.......................................................................... 20
2.3 In-situ Annealed and External Annealed BaM Thin Films Made by RF Magnetron
Sputtering..................................................................................................................................... 22
2.4 BaM thin films Made by the alternating temperature multilayered method............ 25
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2.5 BaM Thick Films Grown by vacuum Liquid PhaseEpitaxy (LPE) and Reflow
M ethod........................................................................................................................................ 27
2.6 Magneto-Optical Kerr Effect (M OKE)......................................................................... 30
2.7 Vibrating Sample Magnetometer (VSM)...................................................
33
2.8 The Hysteresis L oop ........................................................................................................34
CHAPTER 3 ................................................................................................................................37
Results and D iscussion...............................................................................................................37
3.1 External Annealed BaM Thin Films Made by Sputtering..........................................37
3.2 In-Situ BaM Thin Films Made by Sputtering.............................................................. 46
3.3 BaM Films Made by Sputtering on MgO (111) Substrate........................................ 52
3.4 BaM Films Made by Alternating Temperature Multilayered Technique on AI2 O 3
(0001) Substrate...........................................................................................................................55
3.5 BaM Thick Films Made by RF Sputtering................................... ..............................77
3.6 BaM Thick Films Made by vacuum LPE and Reflow M ethod...................
3.7 BaM Thick Films Made by Using BaM Thin Film Seed L ay er
78
...................... 82
CHAPTER 4 .............................................................................................................................. 87
Conclusions and Future W o rk ......................................................................................
87
Bibliography................................................................................................................................89
Appendix A: Landau - Lifshitz - Gilbert - Equation............................................................ 92
Appendix B: Demagnetizing Interaction M atrix........................... ...................................... 100
Appendix C: Expansion of the LLG Equation...............................................
112
Appendix D: Approximation o f Demagnetizing Field........................................................ 114
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List of Figures
Figure 1.1: Cross section o f the magnetoplumbite structure M, with c axis vertical. The
arrows indicate the spin orientations. The drawn vertical lines are axes o f three fold
symmetry. A cross indicates a centre o f symmetry. All layers containing barium are
mirror planes, and are denoted by m ......................................................................................... 1 1
Figure 1.2 (a): Three metal atoms surrounding oxygen atom P .......................................... 13
Figure 1.2 (b): The principal super-exchange interactions in BaFenO jg...........................13
Figure 1.3: The Y-junction circulator......................................................................................15
Figure 1.4: Y-junction circulator operation............................................................................17
Figure 1.5: Resonance curves show basic regions o f circulator operation.........................18
Figure 2.1: BaM target fabrication procedure........................................................................ 20
Figure 2.2 (a): The magnetron: in planar magnetron sputtering arrangement...................21
Figure 2.2 (b): cross section......................................................................................
21
Figure 2.2 (c): the electrons drift in the - E x B direction, actually executing a cycloidal
p ath .........................................................................................................................................
21
Figure 2.3: UI RF sputtering system....................................................................................... 22
Figure 2.4: The BaM thin / thick films using RF magnetron sputtering process.............. 24
Figure 2.5: 0.30 pm, 0.45 pm, and 0.75 pm BaM films made by the alternating
temperature multilayered technique..........................................................................................25
Figure 2.6: BaM thin films using the alternating temperature multilayered technique.. .26
Figure 2.7: The vacuum LPE system...................................................................................... 29
Figure 2.8: The magneto-optical Kerr effect (MOKE) can occur in three different
geometries.................................................................................................................................... 30
Figure 2.9: The experimental layout. (The inset illustrates the arrangement o f the magnet
in longitudinal direction)...........................................................................
32
Figure 2.10: The vibrating sample magnetometer (VSM )....................................................33
Figure 2.11: The hysteresis loop.............................................................................................. 35
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IX
Figure 3.1: The hysteresis loop for the external annealed 0.15pm thin film on Si (100)
using target # 1. The square symbols correspond to an out-of-plane VSM measurement,
and the solid line to an in-plane VSM measurement............................................................. 38
Figure 3.2: XRD o f the external annealed 0.15 pm BaM thin film on Si (100).......
39
Figure 3.3: The EDX (taken from the point 1) o f the 0.15 pm BaM thin film ..................40
Figure 3.4: SEM observation o f the 0.15 pm BaM thin film........................................
41
Figure 3.5: The hysteresis loop for the external annealed 0.15pm thin film on Si (100)
using target # 2. The square symbols correspond to an out-of-plane VSM measurement,
and the solid line to an in-plane VSM measurement............................................................. 42
Figure 3.6: XRD o f the external annealed 0.15 pm BaM thin film made from target # 2
on Si (100)....................................................................................................................................43
Figure 3.7: SEM observation o f the 0.15 pm BaM thin film made from target # 2 .........44
Figure 3.8 (a): Coercivity vs. Thickness dependence o f the BaM external annealed films
........................................................
45
Figure 3.8 (b): Squareness vs. Thickness dependence o f the BaM external annealed
films..........................................
.45
Figure 3.9: The hysteresis loop o f the in-situ annealed 0.15pm thin film on Si (100)
using target # 2. The square symbols correspond to an out-of-plane VSM measurement,
and the solid line to an in-plane VSM measurement........................................................... ..47
Figure 3.10: XRD o f the in-situ annealed 0.15 pm BaM thin film on Si (100)...............48
Figure 3.11: The perpendicular MOKE hysteresis loop of the in-situ annealed 0.15 pm
BaM thin film on Si (100)..........................................................................................................49
Figure 3.12: The EDX (taken at the center) o f the 0.15 pm in-situ annealed BaM film..50
Figure 3.13: SEM observation o f the in-situ annealed 0.15 pm BaM thin film on
Si(100)...........................................................................................................................................51
Figure 3.14: The hysteresis loop o f the 0.15pm thin film on MgO (111). The square
symbols correspond to an out-of-plane VSM measurement, and the solid line to an in­
plane VSM measurement........................................................................................................... 53
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X
Figure 3.15: XRD o f the 0.15 pm BaM thin film on MgO (111)...........................
54
Figure 3.16: SEM observation o f the 0.15 pm BaM film on M g O ( l l l )
55
..........
Figure 3.17 (a): Schematic diagram showing the matching o f rhombohedral and
hexagonal lattices, Hexagon before the deformation............................................................. 56
Figure 3.17 (b): one way o f deforming the hexagon in order to fit it on the rhombohedral
lattice. The area change for this case is -25.7 A2....................................................................56
Figure 3.17 (c): Another way o f fitting the hexagon on the rhombohedral lattice. This
configuration gives an area change o f 3.58A2........................................................................ 56
Figure 3.18: The hysteresis loop o f the 0.30pm thin film on AI2 O3 (0001). The square
symbols correspond to an out-of-plane VSM measurement, and the solid line to an in­
plane VSM measurement........................................................................................................... 58
Figure 3.19: The Simulated Hysteresis loops for the 0.30pm thin film on AI2O3
(0001)..............................................................
59
Figure 3.20: Layers o f A represent layer R and R*, layers of B represent layer S and S*
Figure 3.21: XRD o f the 0.30 pm BaM thin film on AI2 O 3 (0001).......... ...........
.62
Figure 3.22: SEM observation o f the 0.30 pm BaM film on AI2O3 (0001).......
.63
points on the surface o f the 0.30 pm BaM film on AI2O3 (0001)
64
Figure 3.24: The EDX data taken for the surface image o f figure 3.23...............
65
Figure 3.25 (a): Barium EDX m apping.................................................... ............
.66
Figure 3.23:
6
Figure 3.25 (b): Fe EDX mapping o f a selected area o f the 0.30 pm BaM film
surface............................................................................................................................
66
Figure 3.26: The hysteresis loop o f the 0.45pm thin film on AI2O3 (0001). The square
symbols correspond to an out-of-plane VSM measurement, and the solid line to an in­
plane VSM measurement........................................................................................................... 67
Figure 3.27: XRD o f the 0.45 pm BaM thin film on AI2O3 (0001).................................... 69
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XI
Figure 3.28: The hysteresis loop o f the 0.7pm thin film on AI2 O3 (0001). The square
symbols correspond to an out-of-plane VSM measurement, and the solid line to an in­
plane VSM measurement
.............................................................................................70
Figure 3.29: XRD o f the 0.70 pm BaM thin film on A12 0 3 (0001).................
70
Figure 3.30: The hysteresis loop o f the 0.7pm thin film on AI2O3 (0001) using one
deposition layer. The square symbols correspond to an out-of-plane VSM measurement,
and the solid line to an in-plane VSM measurement..........................
72
Figure 3.31: SEM observation o f the 0.70 pm BaM film on AI2 O 3 (0001)....................... 73
Figure 3.32: 4 points on the surface o f the 0.70 pm BaM film on AI2 O 3 (0 0 0 1 )..........7 4
Figure 3.33: The EDX data taken for the surface image o f figure 3.31.............................75
Figure 3.34 (a): Barium EDX mapping.................................................................................. 76
Figure 3.34 (b): Fe EDX mapping o f a selected area o f the 0.70 pm BaM film
surface..........................................................................................
76
Figure 3.35: The hysteresis loop o f the 14pm thick film on Si (100). The square symbols
correspond to an out-of-plane VSM measurement, and the solid line to an in-plane VSM
measurement................................................................................................................................78
Figure 3.36: The hysteresis loop o f the 350pm thick film on (0001) sapphire AI2 O 3
substrate. The square symbols correspond to an out-of-plane VSM measurement, and the
solid line to an in-plane VSM measurement............................................................................79
Figure 3.37: XRD pattern o f the 350 pm BaM thick film on (0001) sapphire AI2 O 3
substrate...............................................................................................................
.80
Figure 3.38 (a): SEM surface image o f the 350 pm BaM thick film on sapphire (A12 0 3)
(0 0 0 1 ) substrate...............................................................................................
81
Figure 3.38 (b): SEM cross section image o f the 350 pm BaM thick film....................... 82
Figure 3.39: The hysteresis loop o f the 250pm BaM thick film using BaM thin film as
seen layer. The square symbols correspond to an out-of-plane VSM measurement, and
the solid line to an in-plane VSM measurement..................................................................... 83
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Figure 3.40: The BaM thick films grown by using seed layer dependence on the
reflowing temperature. The square symbols correspond to an out-of-plane coercivity and
the solid line to in-plane coercivity
............................................................................... 84
Figure 3.41 (a): SEM cross section image o f the BaM thick film using seed layer......... 85
Figure 3.41 (b): SEM surface image o f the BaM thick film using seed layer................... 8 6
Figure A l: Top and side views o f geometric arrangement of the hexagonal cells........... 92
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1
CHAPTER 1
Introduction
Magnetite (Fe3 0 4 ) was the first magnetic material known to man. It was also known
that a shaped piece of magnetite floating on water would turn until it points
approximately north and south, thus was the mariner’s compass born. Research on
magnetic materials can be said to date from the invention of the electromagnet, which
provided much powerful magnetic fields. Many new magnetic materials have been
developed ever since, which paved the road for many modem applications such as the
wireless telecommunication systems, microwave and navigation devices.
Due to their large magnetic anisotropy perpendicular to the film plane, barium
ferrite thick films (BaFei 2 0 i9 , or BaM) with c-axis orientation (the direction of the easy
magnetization vector is perpendicular to the film plane) are attractive candidates for
microwave applications. Circulators, isolaters and phase shifters are some microwave
devices that employ ferrite materials. Modern military communications systems require
these devices to be highly portable and integrated into small chip, which requires the
ferrite material to be in the form of thin or thick film instead of the bulk ferrite. In current
technology, microwave ferrite devices use bulk BaM under an external magnetic field
bias to control the microwave propagation in the circulator. Our research task was set to
fabricate a self-biased microwave circulator, which does not require an external magnetic
field; therefore, the self-biased circulator will be much lighter and smaller in size. To
develop the future generation of microwave devices that can be fully integrated into a
chip, excellent c-axis orientation with moderate coercivity and good squareness are
required so that BaM films can be self-biased. To have a functional circulator operating
in the GHz range, one needs to have a minimum thickness of the magnetic media of 100
micrometers or more. In this chapter we will briefly review the magnetism of materials,
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2
design and theory of operation of the Y junction circulator, then we will talk
about the ferrites and barium hexaferrite (BaFei 2 0 i 9 or BaM). In chapter two we will
explain the experimental set up and the characterization methods that have been used to
grow, and study the magnetic and the structural properties of the thin/thick films. In
chapter three we will show and discuss the results of the thin/thick BaM films. Fihally,
conclusions and future work are presented in chapter four.
1.1 Magnetism in Materials
Magnetism in materials is due to the magnetic dipole moments within the ferrite.
The magnetic properties in which we are interested are due entirely to the electrons of the
atoms, which have magnetic moment by virtue of their motion. There are two kinds of
electron motion: orbital and spin and each has a magnetic moment associated with it.
We can think of the orbital motion of the electron around the nucleus as a current in
a loop of wire having no resistance. With the help of Bohr theory of the atom, the
magnetic moment of the electron in the first (n = 1) Bohr orbit is given by
p = e h / 4 7 tmc
(1.1)
where h is Blank’s constant = 6.62 x 10' 27 erg. sec, e is the charge of the electron = 4.80
x 10' 10 esu, m is the mass of the electron = 9.11 x 10' 28 g, and c is the speed of light = 3.0
x 10 10 cm/sec. The quantity in equation 1.1 has special name called Bohr Magneton and
equals to pe= 0.927x 10' 20 erg/Oe.
Another source of magnetism in matter is the electron spin. The existence of
electron spin was postulated in 1925. The spin is intrinsic property of a particle
independent of its motion. It is found that the magnetic moment due to electron spin is
equal to that due to motion in the first Bohr orbit. The magnetic moment of the atom is
the vector sums of all its electronic moments, which comes from orbital and spin
contributions. If the magnetic moments of all electrons cancel one another out, then the
atom has no magnetic moment. If the cancellation of the moments is partial, then the
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3
atom has net magnetic moment.
1.2 Diamagnetism
A diamagnetic material is a material which exhibits negative magnetism, even
though it has no net magnetic moment. Since the motion of electrons is equivalent to a
current in a loop, and, as soon as the magnetic field is increased from zero to certain
value, the change in flux through the loop induces an electromotive force (emf) in the
loop according to Faraday’s law. This emf acts in such a way as to oppose the change in
flux (Lenz’s law), that is, to decrease it. Therefore, diamagnetic material reacts to the
external field by creating induced small magnetic moments opposing the applied field.
The change in the induced moment of the diamagnetic material A p caused by the applied
field is given by
A p (per electron) = -e2 R 2 H / 6 mc 2
(1.2)
where R is the radius of an orbit that can take place on all possible orientations with
respect to the field H. Atoms of closed shells usually have their electron’s spin and orbital
moments so oriented that the atom has no net magnetic moment. Monatomic rare gases
like He, Ne, etc. have closed shell structures and are all diamagnetic. Also gases like H 2
N 2 , etc. have closed shell structure which means their molecules have no net magnetic
moments [ 1 ].
1.3 Paramagnetism
The importance of paramgnetism theory originates from the fact that it leads
naturally into the theory of ferromagnetism and ferrimagnetism. Unlike diamagnetism,
paramagnetism exhibit positive susceptibilities. In 1895 P. Curie made the first
systematic measurements of the susceptibility of a large number of substances over an
extended range of temperature. The mass susceptibility %was found to be independent of
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4
temperature for diamagnetics, but it varied inversely with the absolute temperature for
paramgnetics according to the following equation
X=
C/T
(1.3)
where T is the temperature, C is curie’s constant and to be defined later in this section.
Curie’s measurements on paramagnetics were not explained theoretically until Langevin
solved the problem in 1905. Langevin found that the magnetization M varies with T and
H as follows
(M / M„) = coth (p H/ KT) - 1/ (p H/ KT)
(1.4)
where M 0 is the maximum possible magnetization the material can have, K is boltzman
factor = 1.38 x 10' 16 erg /deg. For large values of T we find equation 1.4 leads to equation
1.3, and we can find an expression for %as follows
X = N p 2 / 3AKT = C / T
(1.5)
where N is Avogadro’s number = 6.02 x 1023, C = N p 2 / 3AK, A is the atomic weight.
Many paramagnetic materials, however, do not obey Curie’s law (equation 1.3);
they obey instead the more general Curie-Weiss law
X = C / (T-0)
(1.6)
where 0 is constant to be defined later. P. Weiss pointed out this behavior could be
understood by assuming that the moments did interact with one another. He suggested
that this interaction could be expressed in terms of fictitious internal field which he called
the “Molecular field” HmWeiss assumed that the intensity of the molecular field was
directly proportional to magnetization as follows
Hm = y M
(1.7)
where y is called the molecular field constant, the susceptibility is given by
X=
C / (T- 0)
(1.8)
where 0 = pCy, and p is the density of the material
In an atom composed of many electrons, the total angular momentum number J is
given by
J=L+S
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(1.9)
5
where L is the orbital angular momentum number, and S is the spin momentum number
the total magnetic moment peff is given by
Peff111 g Pb (JCJ+l) ) 1' 2
( 1 .1 0 )
where g is called spectroscopic splitting factor [ 1 ].
1.4 Ferromagnetism
There is a specific temperature where the magnetic properties of the material are
switched from one state to another, that temperature is called Curie temperature Tc
(equation 1.3). If the temperature of the material is below Tc the material will be
ferromagnetic, and above that it will be paramagnetic. One important aspect about
ferromagnetic materials is that they could be self saturated or spontaneously magnetized.
This is due to the existence of the molecular field postulated by P. Weiss; this internal
field is so strong that it could magnetize the material into saturation even in the absence
of an applied field. Weiss also postulated that a ferromagnetic in the demagnetized state
is divided into a number of small regions called domains. Each domain is spontaneously
magnetized to the saturation value M s, but the directions of magnetization of the various
domains are such that the specimen as a whole has no net magnetization. The process of
magnetization is then one of converting the specimen from a multi-domain state into one
in which it is a single-domain magnetized in the same direction as the applied field. The
physical origin of the molecular field was not understood until 1928, when Heisenberg
showed it was caused by quantum-mechanical exchange forces. It is this exchange
interaction between the spins which gives rise to ferromagnetism. The exchange energy
forms an important part of the total energy of many molecules and of the covalent bonds
in many solids. If two atoms i and j have spin angular momentum S, h / 2 n and Sj h / 2 n
respectively, then the exchange energy between them is given by
Eex = -2 Jex Sj Sj COS 0
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(1.11)
6
here Jex is a particular integral called the exchange integral, and
0
is the angle between the
spins. The molecular field is given by
Hm = 2 Z J exS2/ p H
(1.12)
where Z is the atom nearest neighbors. Here pn is the magnetic moment in the direction
of the field, also Jex is given by
Jex = (3 K 0) / (2 Z S (S+l))
(1.13)
1.5 Antiferromagnetism
Antiferromagnetic substances have a small positive susceptibility at all
temperatures, as the temperature decreases, %increases but finally go through a maximum
at critical temperature T n called Neel temperature. The substance is paramagnetic above
T n and antiferromagnetic below it. TNcommonly lies far below room temperature, so that
it is often necessary to carry susceptibility measurements down to quite low temperatures
to discover if a given substance, paramagnetic at room temperature, is actually
antiferromagnetic at some lower temperature. Most, but not all, antiferromagnetics are
ionic compounds, namely, oxides, sulphides, and the like.
Just as in ferromagnetism, the clue behavior of antiferromagnetic lies in the way its
susceptibility varies with temperature above the critical temperature. The susceptibility is
given by
X=
C / (T+ 0)
(1.8)
where 0 = pCy, and p is the density of the material. In other words, the material obeys a
Curie-Weiss law but with a negative value of 0. Inasmuch as 0 is proportional to the
molecular field coefficient y, the molecular field Hm, in the paramagnetic region, is
opposed to the applied field H; whereas H tries to align the ionic moments, Hm acts to
disalign them. The result is that any tendency of a particular ionic moment to point out in
one direction is immediately counteracted by a tendency for the moment on an adjacent
ion to point in the opposite direction.
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7
Below the critical temperature Tn, this tendency toward an antiparallel alignment
of moments is strong enough to act even in the absence of an applied field, The lattice of
magnetic ions in the crystal then breaks up into two sublattices, designated A and B,
having moments more or less opposed. The tendency toward antiparallelism becomes
stronger, the lower the temperature is below Tn, until at 0°K the antiparallel arrangement
is perfect. We now see that an antiferromagnetic at 0°K consists of two interpenetrating
and identical sublattices of magnetic ions, each spontaneously magnetized to saturation in
zero applied field, but in opposite directions, just as the single lattice of a ferromagnetic
is spontaneously magnetized. Evidently, an antiferromagnetic has no net spontaneous
moment and can acquire a moment only when a strong field is applied to it [ 1 ].
1.6 Ferrimagnetism
Unlike the ferromagnetism, neighboring spins are aligned antiparallel to one
another in antiferromagnetism so that their magnetic moments cancel each other out.
Neel assumed that the antiferromagnetic materials consisted of two sublattices having
moments opposed. Ferrimagnetism can be thought of as imperfect antiferromagnetism in
which the opposed magnetization in the sublattices are not equal, and a net spontaneous
magnetization results. Their spontaneous magnetization disappears above a certain
critical temperature Tc, also called the Curie temperature and then they become
paramagnetic.
The most important ferrimagnetic substances are certain double oxides of iron and
another metal called ferrites. Magnetite (Fe 3 0 4) for example is an oxide compound which
is known as a spinel ferrite. In the next sections I will review the crystal structure of
different ferrites. The magnetic ferrites fall mainly into two groups with different crystal
structures: cubic and hexagonal structures. Cubic ferrites contain spinel and garnet
ferrites [2 ].
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8
1.6.1 Spinel Ferrites
The general formula of spinel ferrites is MeFe 2 C>4 , where Me represents a divalent
metal ion (cation) with an ionic radius approximately between 0.6 and 1A°. In the case of
simple ferrites, Me is one of the divalent ions of the transition elements Mn, Fe, Co, Ni,
Cu and Zn, or Mg and Cd.
The spinel structure takes its name from the mineral MgAI2 O 4 , which crystallizes in
the cubic system. The crystal structure was first determined by Bragg. The unit cell of the
spinel crystal contains eight molecules of MeFe 2 C>4 . The relatively large oxygen ions
form an fee lattice. The unit cell contains 32 oxygen anions. There are 16 trivalent cations
and
8
divalent cations. Two types of crystallographic sites may be occupied by the
cations. One, called the tetrahedral site, or A site, is surrounded by 4 oxygen ions. The
other, called the octahedral site, or B site, is surrounded by
of the normal spinel arrangement requires that the
occupy
8
8
6
oxygen ions. The definition
divalent metal ions of the unit cell
A sites and the 16 trivalent ions occupy the 16 B sites. It has been found,
however, that the magnetic ferrites have a structure known as the inverse spinel in which
8
of the 16 trivalent ions occupy the entire A sites. The other half of the trivalent ions and
all of the divalent ions occupy the B sites in random order. Experiments showed that
magnetite Fe 3 (> 4 and nickel ferrite NiFe 2 C>4 were inverse spinel. Contrast to zinc ferrite
ZnFe2 C>4 it was found to be a normal spinel structure [3].
The magnetic moment on a A site is oriented antiparallel to one on B site due to the
strong A-B interaction. Hence, the total magnetic moment is given by the difference in
magnetic moment of the ions on the two sites. In spinel ferrites, the cations distributions
can be represented by the following
Mex2 +Fe 3 +1.x [Mei.x2 +Fe 3 +1+x] 0
4
(1.14)
where the ions on the tetrahedral sites are given in front of the square brackets and the
octahedral ions between the brackets. For a completely random distribution, x = 1/3, for a
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9
normal spinel x = 1 and for inverse spinel x = 0. The quantity x is the measure of the
inversion [2 ],
1.6.2 Garnet Ferrites
The general formula for the magnetic garnets is M 3 Fe 2 (F e O ^ , where M may be
nonmagnetic trivalent yttrium or lutetium, M can also be one of the magnetic rare earth
ions like La, Ce, Pr, d, Pm, Sm, Eu, Gd, Tb, and Yb. If the M ions are nonmagnetic, the
iron ions form two nonidentical antiparallel sublattices. If M ions are rare earths having a
permanent magnetic moment, they form a third magnetic sublattice which causes the
substance to have a compensation point below room temperature. The Neel temperature
of garnets is always above 275°C.
There are
8
of the formula units MsFe2 (FeCDs in the cubic unit cell. For more
details on the structure of the garnet ferrite, B. Lax’s book can be consulted [3]. In some
respects the ferrimagnetic oxides having the crystal structure isomorphic with the
classical garnet Ca 3Fe 2 ( S iO ^ are superior to ferrites. This is particularly true for one
member of the class Y 3Fe 2 (FeCLL. Since its discovery in 1956, it has proved to be of
great importance both in high frequency applications and for basic research experiments
on ferromagnetic resonance. The crystal structure is very nearly cubic and contains
essentially no magnetic disorders. The Fe ions, which are responsible for the magnetic
moment, are all trivalent having no orbital angular momentum so that no fluctuating
perturbing fields are present. Having only trivalent magnetic ions, prevent the small
electrical conductivity experienced in ferrites by means of exchange of electrons among
Fe2+ and Fe3+ ions and provides high resistivity value. The smallest ferromagnetic line
width on a carefully polished yttrium garnet sphere is 0.6 oersted at 3 cm wavelength at
room temperature. The saturation magnetization at room temperature Ms is rather low 4n
M s= 1700 Gauss.
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10
1.6.3 Hexagonal Ferrites
The second group of magnetic ferrites is the hexagonal ferrites; they have
hexagonal crystal structure and also called hexaferrites. In this dissertation, our main
discussion of ferrites is focused on barium hexaferrites. Barium hexaferrite has a crystal
structure which, according to Adelskold [4], is equal to that of the mineral
magnetoplumbite, the composition of which is approximately PbFe 7 .5Mn 3 .5 Alo.5 Tio.5 O 19 .
The ferrimagnetic magnetoplumbite (also called M type) hexaferrite has the general
formula P 0 .6 (Fe 2 0 3 ), in which PO may replaced by Ba2+, Sr2+, or Pb2+. The barium
hexaferrite is one member of the M type family, which has the chemical formula
BaFenOio. Besides M type, there are other hexagonal hexaferrite groups like W, Y, and
X type hexaferrites. B aF en O ^is of considerable practical importance because it exhibits
high resistivity, has an internal uniaxial anisotropy field HA= 17000 oersteds, has a high
saturation magnetization
4 7 tMs,
and is a permanent magnet with a coercive force of more
than 3000 oersteds. The crystal exhibits its high magnetic anisotropy along only one axis
of easy magnetization, resulting in the large coercive force.
Figure 1.1 shows the barium ferrite unit cell cross section view. The crystal
structure of the BaM is uniaxial structure, because the oxygen anions are positioned in as
hexagonal closed packed, so that its layer sequence is perpendicular to the [0 0 1 ]
direction. The length of the c axis corresponding to this structure is 23.2A and that of a
0
«
axis is 5.88A . In an elementary cell each layer contains four large ions. There are in four
successive layers always four oxygen ions, but each fifth layer contains three oxygen ions
and one barium ion. The magnetoplumbite structure can be build up from spinel blocks of
two oxygen layers being blocks S and S*, which are connected by a block R containing
the barium ion. Blocks R* and S* are obtained from blocks R and S, respectively, by
rotation over 180° around the c axis. The layer containing barium is hexagonally packed
with respect to two oxygen layers at each side. The four oxygen layers between those
containing barium are cubically packed.
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Figure 1.1: Cross section of the magnetoplumbite structure M, with c axis vertical. The
arrows indicate the spin orientations. The drawn vertical lines are axes of three fold
symmetry. A cross indicates a centre of symmetry. All layers containing barium are
mirror planes, and are denoted by m [2 ].
There is an overlap between cubically and hexagonally packed sections in the
structure. The basal plane containing the barium is a mirror plane of the R block, and
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12
consequently the blocks preceding and succeeding the R block (S and S*) must be rotated
over 180° with respect to each other. In general it can be said that when one R block is
passed in a structure the following blocks must be rotated over by 180° around the c axis.
Only after a second R block is passed is the original situation found again. The
crystallographic structure can thus be described as RSR*S*, and the unit cell contains a
number of ions corresponding to 2 (BaFei 2 0 i 9 ). The unit S is, then, contains two
molecules Fe 3 0 4 . For the ferric ions there are three different kinds of interstitial sites
present. Along with octahedral and tetrahedral sites there exists a new type of interstitial
site which is not found with spinels and which is surrounded by five oxygen ions
constituting a triagonal bipyramid. These sites occur in the same layer as barium ion, and
they can be compared with tetrahedral sites. In the hexagonal structure two tetrahedral
sites are adjacent to each other and for these two only one metal ion is available. This
metal ion now occupies a position halfway between them, amidst the three oxygen ions
[2],
The superexchange interactions for Fe 3+-0 -F e3+ in magnetoplumbites are strong and
negative when the angle through the intervening oxygen ion is near 180°. In Figure 1.2
(a) below, three metal ions surrounding oxygen atom P, where the angle Fe 3 +(l)-0 Fe 3+(2) is 140° and the angle Fe 3 +(2)-0-Fe 3 +(3) is 80°. Therefore, the superexchange
interaction between Nos. 2 and 3 is negligible and they both align antiparallel to No. 1.
Ion 3, in turn, is coupled to the S4 block by the strong negative interaction Fe 3 +(3)-0Fe 3 +(4) as shown in figure 1.2 (b). Therefore, the alignment of ions 3 and 4 are
antiparallel. In this later case, it is the fact that the ions are so near to the intervening
oxygen ion that is the determining factor. The interaction Fe 3 +(l)-0 -F e 3+(4) does not
compete effectively because the distance of N o.l from the intervening oxygen ion is
about 2.3A compared with the typical interaction distance of about 1.3 A . The above
description in the vicinity of the barium layer (R layer) accounts for the uniaxial
anisotropy for barium ferrite along c axis. Since the cubic spinel block has no strongly
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preferred axis of alignment, the coupling from the barium layer imposes its strong
preferential orientation on the spinel block [3].
Figure 1.2 (a): Three metal atoms surrounding oxygen atom P [3].
Figure 1.2 (b): The principal super-exchange interactions in BaFe^Oig [3].
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14
Each Fe3+ ion contributes 5[Xb to the magnetic moment at absolute zero, and we may
from an algebraic sum of the contributions to determine the saturation magnetization.
Therefore, the resultant magnetic moment per formula unit BaFei 2 0 i9 is equal to the sum
of the moments of the seven octahedral ions and the ion in the layer containing barium,
reduced by the moments of two octahedral and two tetrahedral ions, which are oppositely
oriented to them; this is schematically indicated by arrows in fig tu rel.l. The
magnetization per formula unit is equal t o ( l + 7 - 2 - 2 ) x 5 = 20pB. Measurements on
polycrystalline BaFei 2 0 i 9 at liquid hydrogen temperature and in fields up to 26,000
oersteds result in exactly this value, at about 20°C the saturation magnetization
4 7 tMs is
about 4775 Gauss. Curie’s temperature for BaFei 2 0 i9 is 450°C [2].
In 1929 the Russian physicist Akulov showed that the anisotropy energy E can be
expressed in terms of series expansion of the direction cosines of Ms relative to the
crystal axes. Generally, in a cubic crystal, if Ms makes angles a, b, c with the crystal axes,
and ai, 012,013 are the cosines of these angles then E can be expressed as
E = Ko+ Ki (di^Cl2 ^ + CL^CL^ + (Xi^Ot.3 ^) + K 2 (CL\^0.^ Of?2)
(1-15)
where Ko, Ki, K 2 are constants for a particular material, K 2 is sometimes small so it can
be neglected, K 0 is independent of angle and usually is ignored, and when K 2 is zero, the
direction of easy magnetization is determined by the sign of Ki. For BaM, the direction
of easy magnetization is the hexagonal c axis; in this case, the anisotropy energy E
depends on only a single angle, the angle 0 between the Ms vector and the c axis.
Therefore, E for the hexagonal structure can be expressed as
E = K 0 + K 1 cos20 + K 2 cos40 + ...
(1-16)
It is common to write the last equation for E in hexagonal crystals in powers of sin0
substituting cos2 0 = 1 - sin 2 0 into the equation 1-16, we have
E = K0+ Ki sin20 + K 2 sin40 + ...
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(1-17)
15
when Ki is positive and K 2 > -Ki, the energy E is minimum for 0 = 0, and the c axis is
one of easy magnetization. These conditions are met for cobalt and BaFe^Ow, a crystal
with single easy axis, along which the magnetization can point either up or down, is
called a uniaxial crystal [ 1 ].
1.7 The Y-Junction Circulator
A circulator is an element for giving non-reciprocal characteristics to a high
frequency circuit so as to rout power to a specific location within a circuit. Many modem
military applications and portable or mobile communication equipments use circulators.
In figure 1.3, we show the Y -junction circulator
Copper Trace
Dielectric
Copper Ground Plane
Figure 1.3: The Y-junction circulator [5].
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16
The Y-junction circulator uses a permanent magnets to bias the ferrimagnetic disks.
If a self biased ferrimagnetic can be obtained, the permanent magnets can be excluded
from the Y-junction circulator which makes it lighter and smaller in size. A ferrite disk
and the intersection of 3 transmission lines form the Y-junction where the actual
circulation occurs. The ferrite material is essentially enclosed in a cavity, forming a
cylindrical cavity resonator. When power is applied from any of the three transmission
lines, a complex wave pattern is established. This electromagnetic field pattern is due to
counter-rotating waves.
The presence of an axial magnetic field across the ferrite material changes the
effective permeability seen by the rotating waves, but in a direction that depends on the
sense of rotation. This will create a two-counter rotating waves with different phase
velocities vp as illustrated in figure 1.4. The goal is to create a constructive interference at
the non-isolated port, and a destructive interference at the isolated port. In practice,
circuit tuning can be used for some adjustment to the amount of rotation of the standing
wave pattern [5].
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17
Destructive
interference
Constructive
interference
\
Saturation
O
Magnetization
Figure 1.4: Y-junction circulator operation [5].
When a ferrite material is magnetized the magnetic moments of the electrons
precess (wobble like a spinning top) at a frequency proportional to the biasing magnetic
field. Ferrimagnetic resonance occurs when a rotating RF magnetic field has the same
direction and frequency as the precessing electrons in the ferrite material. The maximum
coupling of the energy from the RF signal to the ferrite material will occur at
ferrimagnetic resonance. If the direction of rotation or the frequency of the RF signal is
changed, minimum coupling will occur. A simplistic analogy can be used to explain this
phenomenon. It is easier for a person to pass items to an individual riding on a merry-goround if he is running in the same direction and at the same speed while it is more
difficult to pass them if both are moving in opposite directions. However, biasing the
junction circulator at ferrimagnetic resonance is not desirable because the device would
be extremely lossy. High insertion loss can also occur at very low biasing magnetic fields
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18
this low field loss region arises from the fact that the applied magnetic field is not
sufficient to fully saturate or align the individual magnetic domains of the ferrite material.
Although high loss occurs in both the low field and ferrimagnetic resonance areas low
loss operation can still be obtained in the below and above resonance regions as shown in
figure 1.5.
i
ti§ s
BilSSftETItJ FIELD
Low Field
L ass fiiffipr?
f ert® ajf«tlic
f if ia iin c t ;
|
Above Resonance
Operating Region
Figure 1.5: Resonance curves show basic regions of circulator operation [5].
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19
CHAPTER 2
Experimental Procedure
In this chapter, the experimental procedure for depositing barium ferrite films is
described. We have used the RF magnetron sputtering to grow our thin films. To
crystallize the thin films, we have used two ways; the first is to anneal the films in the air,
the second, is by using in-situ annealing. For barium ferrite thick films, we have used
both the liquid phase epitaxy method (LPE) and the RF magnetron sputtering. Many
characterization methods have been used to study the structural, magnetic, and surface
properties of the films. X-ray diffraction (XRD) was used to study the crystallographic
structure of the BaM films. The vibrating sample magnetometer (VSM) was employed to
investigate the magnetic properties of the films, and magneto-optic kerr effect (MOKE)
was used to study the surface magnetic properties. Scanning electron microscope (SEM)
was used to explore the BaM film’s surface and to measure the thickness of the films by
the cross section images and electron dispersive spectroscopy (EDX) was used to check
the atomic ratio of the BaM thin films.
2.1 Making of BaM Target
We have made two enriched barium ferrite targets so the barium to iron ratio of
the films can be fixed; the first one (Target # 1 ) was made of 50.0 g commercial
BaFei 2 0 i9 powder and 6.617 g of BaCC>3 , the second one (Target # 2) was made of 50.0
g commercial BaFe^Oig powder and 7.617 g of BaCC>3 . Both targets were made by the
same solid state reaction method. After mixing the powder thoroughly, they were
grounded in mortar for 4 hours, and then heated for 2 hours at 830°C in air. The powder
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20
then was cold pressed to the shape of a disk 2.0 inch diameter. The final stage was to heat
the BaM disks as shown in figure 2.1.
Heating from room temperature to 300°C in 1 hour.
Keeping the temperature at 300°C for 1 hour.
Increasing the temperature to 1000°C in 2 hours
and 2 0 minutes.
Keeping the temperature at 1000°C for 3 hours.
Increasing the temperature to 1200°C in 2 hours
and 30 minutes.
Keeping the temperature at 1200°C for 6 hours.
Cooling the temperature down to
temperature in 5 hours and 25 minutes.
room
Figure 2.1: BaM target fabrication procedure.
2.2 The RF Magnetron Sputtering
Generally, BaM thin films can be deposited by various vacuum deposition
techniques such as evaporation [6 ], sol-gel technique [7], pulse laser deposition (PLD) [8 ,
9], and RF sputtering [10, 11, 12, 13, 14]. RF sputtering has become an important
technique for depositing homogeneous BaFe^Oig films. High-energy electrons in a low
pressure environment can transform a normally inert gas like Argon into highly reactive
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21
positive ions Ar+. Generally, a target (the cathode of the discharge), is bombarded by the
positive ions. Target atoms are ejected and impinge on a substrate (the anode of the
discharge) forming a coating. The RF magnetron which shown in figure 2.2(a, b) is a
magnetically assisted discharge. The power supply is a high voltage RF source which
provides an electric field perpendicular to the target surface. A permanent magnet is
added to create lines of magnetic flux that are perpendicular to the electric field and thus
parallel to the surface of the target. The electrons near the target surface will be trapped.
Trapping of the electrons occur because of the drift velocity in the - E x B direction, but
superposed on this velocity is a cycloidal motion depicted in figure 2.2 (c). The radius of
the electrons orbits is given by
r = (m v.1.) / (q B)
(2 . 1)
where v_i is the component of the electron velocity that is perpendicular to the flux lines,
m is the mass of the electron, q is the charge of the electron and B is the magnetic field.
Generally speaking, the magnetron effect results in enhanced ion bombardment and
sputtering rates, therefore, a magnetron discharge is much efficient with either DC or RF
excitation, than one that does not utilize magnetic trapping.
E
B
Target
(cathode)
Target
r ~ g ' jftm u ta r
|jf$: I-Magnet
M ag n e tic P e t e P ie c a
Figure 2.2: The magnetron: (a) in planar magnetron sputtering arrangement, (b) cross
section, (c) the electrons drift in the - E x B direction, actually executing a cycloidal path.
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22
2.3 In-situ Annealed and External Annealed BaM Thin Films
Made by RF Magnetron Sputtering
Our system is RF magnetron sputtering like the one already described and shown in
figure 2.3. The target is a home made or commercial barium ferrite disk 2.0 inch in
diameter; usually the target is fixed at about 2.5 inch above the substrate.
Figure 2.3: UI RF sputtering system.
Different substrates have been used to deposit BaM thin films, T. L. Hylton et.al.
[15] studied the effect of the substrate on growing BaM thin films. We have chosen
silicon (100), MgO (111) and AI2 O 3 (0001) sapphire substrates to grow our BaM thin film
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23
on. To grow BaM thin film using the RF sputtering we start by pumping the chamber
using a mechanical pump until we reach about 5 x 10~2 torr, then we turn on the
magnetically turbo molecular pump which can reach a pressure of about 3x1 O' 6 torr. A
total pressure of 2.5 x 1O' 2 torr of argon (80%) and oxygen (20%) was introduced into the
chamber. Usually we use 50 W of RF power to start the plasma, during the deposition of
the film, the substrate is kept at fixed temperature, to have good c axis orientation. It was
reported that [10,12,16] Si (100) temperature should be kept above 500°C to grow c-axis
BaM thin films which agrees with our results. Mainly we have found that silicon
substrate temperature should be in the range of 550 - 620°C. We have used ceramic
boraelectric heater to heat up the substrates, and the temperature of the substrate was
measured by Omega thermocouple which was attached to the substrate. Many groups
have studied growing BaM thin films on MgO (111) [17, 18], and on AI2 O 3 (0001)
sapphire substrates [8 , 19]. The temperature of the substrate should be kept at 920 960°C during deposition; our results also agree with that. In our system, it took about 2527 Volt from the variac voltage source to raise the MgO (111) or the AI2O3 (0001)
sapphire substrates temperature to 920 - 960°C. The thickness of the film depends on the
sputtering time; our system growth rate is about 0.1-0.15 pm/h according to VB-250 vase
elliposemeter thickness measurement. The deposition pressure was kept about
8
x 10' 3
torr. After the deposition, films grown on silicon and MgO were amorphous, so in order
to crystallize them they were annealed in air at about 840°C for 10 minutes.
Films grown on silicon also have been crystallized using the in situ method in
which the films were heated inside the chamber without breaking the vacuum. Typically
after deposition, the pump was shut down, 140 torr of argon (80%) and oxygen (2 0 %)
was introduced into the chamber, then immediately increased the substrate temperature
up to 850 - 900°C for 10 minutes. Films grown on sapphire substrates however, were
crystallized right after deposition and no further annealing was necessary. The thin / thick
films preparation procedure using the RF magnetron sputtering is shown in figure 2.4.
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Cleaning the substrate using acetone
and distilled water; substrate is fixed at
about 2.5 inch from target.
Pumping down the chamber to about
5x 1 0 " 06 torr.
Introducing total pressure of 2.5 x 10' 02
torr of Ar + O 2 .
Heating Si substrate at 550 - 620°C.
MgO, and AI2 O 3 substrates are heated at
920 - 960°C.
Starting the magnetron sputtering using
50 W RF signal for different periods of
time.
Crystallizing the thin / thick films by annealing.
Sapphire films do not require annealing. MgO
and Silicon films are annealed in air at about
840°C for 10 minutes.
Crystallizing the thin / thick films by using insitu annealing as follows: pump was shut down,
140 torr of argon (80%) and oxygen (20%) was
introduced into the chamber, then immediately
increased the substrate temperature up to 850 900°C for 10 minutes.
Figure 2.4: The BaM thin / thick films using RF magnetron sputtering process.
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25
2.4 BaM thin films Made by the alternating temperature
multilayered method
BaM thin films were deposited on a clean lOmmxlOmm sapphire (0001) substrate
using an RF magnetron sputtering system. A magnetically turbo molecular pump was
used to reach 3x10' torr. The gas pressure during deposition was fixed to about 8x10'
torr using 20% pure oxygen and 80% pure argon. The target was commercial barium
ferrite 2.0 inch diameter disk and placed at 2.25 inch from the substrate. The RF power
was fixed at 50Wand RF sputtering was on for 90 minutes. During the deposition, the
substrate temperature was kept between 920 - 96Q°C. After the first layer, the plasma
was stopped for about 15-20 minutes, and the substrate’s temperature was reduced to
about 800 - 820°C. The plasma was then resumed again to grow another layer at the
same condition of the first layer. This procedure was repeated until the final thickness
was reached. We grew two, three, and five layers to reach thicknesses of about 0.30 pm,
0.45 pm, and 0.70 pm respectively. The as grown films showed magnetic properties, so
no post annealing was required to crystallize the films, which indicates that the films
were actually in-situ grown. Furthermore, all layers were deposited on bare sapphire
(0 0 0 1 ) substrate without the deposition of seed layer which simplified the deposition of
the films. The thin films preparation procedure made by the alternating temperature
multilayered method is shown in figure 2.5 and 2.6.
0 .1 5 Mm BaM film ,9 5 0 °C
0 .1 6 Mm BaM fllm ,8 2 0 °C
0 .1 5 p m BaM film ,950°C
0 .1 6 Mm BaM film ,9 5 0 °C
0 . 1 6 Mm B a M f il m ,8 2 0 ° C
0 . 1 6 p m B a M f il m ,9 6 0 ° C
a i 2o 3
0 .1 6
|j i t i
BaM film ,8 2 0 °C
0 .1 6 |jm BaM film ,960°C
a i 2o 3
0 .1 6 Mm BaM film ,8 2 0 °C
0 .1 5 Mm BaM fi1m ,950°C
A l20 3
Figure 2.5: 0.30 pm, 0.45 pm, and 0.75 pm BaM films made by the alternating
temperature multilayered technique.
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26
Cleaning the substrate using acetone
and distilled water, substrate is fixed at
about 2.25 inch from target.
Pumping down the chamber to about
5x 10' 06 torr.
Introducing a total pressure of 2.5 x 10' 02
torr of Ar + O 2 .
Heating AI2 O 3 substrate at 920 - 960°C.
Depositing the first BaM layer using 50 W
RF sputtering signal for 90 minutes.
Stopping the plasma for 20 minutes. Reducing
substrate temperature to about 800 - 820°C,
then plasma is resumed using 50 W RF
sputtering signal for 90 minutes.
Repeating this process (the alternating
temperature multilayered technique) two,
three, and five times to reach thicknesses
of about 0.30 pm, 0.45 pm, and 0.7 pm
respectively.
Figure 2.6: BaM thin films using the alternating temperature multilayered technique.
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27
2.5 BaM Thick Films Grown by vacuum Liquid Phase Epitaxy
(LPE) and Reflow Method
W e 1 have realized that even with the optimized sputtering conditions, it is hard to
grow good c-axis oriented BaM thick film using RF sputtering as the c-axis orientation
continues to deteriorate as the thickness increases.
It was reported that LPE is more convenient and efficient process to grow thick
films to the order 50pm or more [12, 17, 20, 21, 22], Most of the previous work on LPE
was performed in air furnaces; these thick films have excellent c-axis orientation and
coercivity of approximately 10 Oe which indicates that they have almost single crystal
structure [17]. W e 1 have built LPE system in two different types: the conventional
furnace LPE (in air) and the vacuum LPE. In both systems; the BaM flux melt was
prepared from a mixture of BaC 0 3 , Fe 2 0 3 and B2O3. The powders were ground and
mixed homogeneously. In the furnace LPE a 120 mL platinum crucible was used to hold
the melt. The starting amount of powder mixture occupied half of the volume of the
crucible. The crucible was then placed in the furnace and heated to about 1200°C. The
melt usually occured after about 2 hours of heating in the furnace. More powder was
added to the crucible during this heating process as the carbonates decomposed into CO2
gas and evaporated.
To ensure the homogneousity of the melt, we stirred the melt for about 8-12 hours.
After getting a homogeneous melt; it was cooled to room temperature.
The idea of vacuum LPE was introduced in order to achieve better decomposing of
carbonates into CO 2 and BaO. Mainly, as the pressure goes down inside the vacuum
chamber, the carbonate environments will be under lower pressure, and that will pull CO 2
out of the carbonates and therefore achieving better quality of the BaM melt. With
vacuum LPE, the same experimental steps which was done in conventional furnace was
1 In collaboration with Prof. David M cllroy and Dr. Yanko Kranov.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
28
done in the vacuum LPE, with the final pressure of the chamber being kept at about 0.3
Torr. Crucible made from quartz was used to melt the powder at about 1200°C; stirrer
speed was about lOrpm. The powder was melted in a quartz crucible attached to a
tungsten coil heater that can hold up to 60 Amps of current. Once we were satisfied with
the uniformity of the melt, it was then slowly cooled down to room temperature under
vacuum. The produced solidified melt at this point was amorphous, i.e., without crystal
I")
orientation. To obtain the preferred c-axis orientation of the BaM melt, we developed the
reflow method in which a small piece was cut out from the solidified molten. The
approximate weights of these pieces were in the range of 0.03 to 0.05 grams. A piece is
of the BaM melt was placed onto a polished sapphire (AI2 O 3 with (0001) orientation)
substrate. This was then placed in a furnace and heated to 1200°C in air for one hour to
remelt the BaM. The actual formation of the preferred c-axis orientation of the barium
ferrite crystals was observed to grow after a slow pace cool down was initiated. Cooling
rates between 3 to 5 degrees Celsius per minute produced the best c-axis orientation of
the films. As a result of the reflow method, a thick BaM film was formed on the
substrate. The thickness of the film was dependent upon the size of the precut solidified
specimen and the size of the substrate. Films with thicknesses of 0.2 to 0.5 mm are
routinely obtained. The vacuum LPE system is shown in figure 2.7.
2 In collaboration with Prof. David M cllroy and Dr. Yanko Kranov.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 2.7: The vacuum LPE system.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
30
2.6 Magneto-Optical Kerr Effect (MOKE)
W e have used the (MOKE) system to study the surface magnetic properties of our
thin films. The magneto-optical kerr effect can be defined as a change in the polarization
state of light after being reflected by a sample in an applied magnetic field. The magneto­
optical effects were first discovered by Michael Faraday (the Faraday effect is observed
in transmission through a material) and J. C. Kerr (the Kerr effect is observed on
reflection from a material). Faraday first discovered his effect in glass rod when a
magnetic field is applied along the direction of propagation of an optical beam. The
intrinsic effects are usually small in such cases. In magnetic media (ferromagnetic or
ferrimagnetic) the effects are much larger, although they are still small and often difficult
to detect. The magneto-optical effect can generally occur in one of three geometries,
which are defined by the orientation of the applied field with respect to the plane of
incidence as shown in figure 2 .8 .
Polar M O KE
Transverse MOKE
longitudinal MOKE
Figure 2.8: The magneto-optical Kerr effect (MOKE) can occur in three different
geometries.
If the magnetization vector is perpendicular to the reflection surface and parallel to
the plane of incidence, the effect called polar kerr effect. In the Longitudinal effect the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
31
magnetization vector is parallel to both the reflection surface and the plane of incidence.
When the magnetization vector is perpendicular to the plane of incidence and parallel to
the surface it is said to be in “Transverse configuration”. For an arbitrary orientation of
the magnetic field, the total magneto-optical effect is a combination of the three specific
effects. All magneto-optical effects are based on the variation of the medium’s refractive
index and the local magnetization of the material. This anisotropy is gyroelectric property
of the material and can be found from the off-diagonal elements of the permittivity tensor
and the electrical susceptibility tensor % =_8 r - 1 of the medium
e
8 =
-jQ e
0
jQ e
e
0
0
0
8
( 2 .2)
In equation 2.2 above, the absolute value of the usually complex magneto-optical
parameter Q is proportional to the magnetization M, which can be caused by an external
field or originate from the spontaneous magnetization of the material itself. The phase Q
is a measure of the change of the ellipticity of either the transmitted or reflected light
wave with respect to the incident light wave. Reversing the direction of magnetization
results in a 180° phase shift of Q. the optical behavior of the material is completely
determined by the real and the imaginary part of 8 and Q. Magneto-optical effects in thin
films can be used for domain structure characterization as well as for utilization of
domain wall dynamics for magneto-optical sensors. In figure 2.9 below we show the
experimental layout, the light source in this experiment was a laser with a wavelength
of 635nm. The light passes through polarizer set to 45°. The Photoelastic Modulator
(PEM) is aligned such that the modulation axis is along the y-axis so that the linearly
polarized light will be incident upon the PEM at an angle of 45° with the modulation axis.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
32
Bipolar ftw ef Supply
DC Voltmeter
Gmssmeter
Lock-In Amp
Rotafabte •
Polarizers
Photoelastic
1.2Tesla |
fledromagnei
Modulator
REM Controller
Figure 2.9: The experimental layout. (The inset illustrates the arrangement of the magnet
in longitudinal direction).
The PEM will add a periodically changing retardation X\ that depends on the
frequency f, which is 50 kHz for this PEM. After the light is reflected from the center of
the electromagnet poles (in polar configuration the light will be reflected from a hole
through one of the poles), it passes through an analyzer which is rotated by an angle
0
with respect to the x-axis. A photodiode is used to detect the light that passes through the
analyzer.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
33
2.7 Vibrating Sample Magnetometer (VSM)
We have used the VSM to study the bulk magnetic properties of our thin / thick
films. Figure 2.10 below shows the schematic of VSM. This method was developed by
Foner [23], and is based on the flux change in a coil when a sample vibrated near it.
Reference
specim en
Loudspeakei
Reference coils'
D etection coils
Sample
Figure 2.10: The vibrating sample magnetometer (VSM).
The sample, usually thin or thick film, is fixed to the end of a rod, the other end of
which is fixed to loudspeaker cone or to some other kind of mechanical vibrator. Current
through the loudspeaker vibrates the rod and sample at about 80 cycles / sec in a direction
at right angles to the magnetic field. The oscillating magnetic field of the sample of the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
34
sample induces an alternating electromotive force (emf) in the detection coils. The
vibrating rod also carries a reference specimen, in the form of a small permanent magnet,
near its upper end; the oscillating field of this induces another (emf) in two reference
coils. The voltages from the two sets of coils are compared, and the ratio is proportional
to the magnetic moment of the sample. The apparatus must be calibrated with a specimen
(spherical shape) of known Ms, this method is very sensitive and versatile. It may be
applied to both weakly and strongly magnetic substances, and it can detect a change in
magnetic moment of 5 x 10' 5 erg/Oe, which corresponds to a change in mass
susceptibility of 5 x 10' 9 emu/g. Oe for one gram sample in a field of 10,000 Oe.
2.8 The Hysteresis Loop
Both ferromagnetic and ferrimagnetic materials differ in the way which they can be
magnetized. If a small applied field suffices to produce saturation, the material is said to
be magnetically soft; if a larger field value is required to saturate the material, then the
material is magnetically hard. Figure 2.11 below shows magnetization curves both in
terms of B (full line from the origin in first quadrant) and M (dashed line). Although M is
constant after saturation is achieved, B continues to increase with H; the relation between
B, H and M (in cgs units) is given by
B = H + 4 tcM
(2.3)
The curve B vs. H from the demagnetized state to saturation is called the initial or
normal induction curve. Sometimes the intrinsic induction, or ferric induction, B, = B - H
is plotted as function of H. Since B - H = 4?rM, such curve will differ from M, H curve
only by a factor of 4n.
If H is reduced to zero after saturation being reached in the positive direction, the
induction B (magnetization M) in ring specimen will decrease from Bs to Br (Ms to M r),
called the retentivity or residual induction. If the applied field is then reversed, the
induction or the magnetization will decrease to zero when the negative applied field
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
35
equals the coercivity Hc. This is the reverse field necessary to coerce the material back to
zero induction.
Figure 2.11: The hysteresis loop.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
36
If the reversed field is further increased, saturation in the reverse direction will be
reached at -Bs. If the field is then reduced to zero and applied in the original direction, the
induction will follow the curve -Bs, -Br, +BS. The loop traced out is known as the major
hysteresis loop, when both tips represent saturation. It is symmetrical about the origin as
a point of inversion, i.e. if the right-hand half of the loop is rotated 180° about H axis, it
will be the mirror image of the left-hand half.
If the process of the initial magnetization is interrupted at some intermediate point
such as a and the corresponding field is reversed and then reapplied, the induction will
travel around the minor hysteresis loop abcdea. Here b is called the remanence and c is
the coercive force..
There is infinite number of symmetrical minor hysteresis loops inside the major
loop, and their tips all lie on the induction curve. There are also many nonsymmetrical
minor loops, some of which are shown at fg and h k [ 1 ].
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
37
CHAPTER 3
Results and Discussion
In this chapter, we will show and discuss the results of the BaM thin and thick films
by using both the RF sputtering and vacuum LPE system. We will then study briefly the
effect of BaM seed layer fabricated by the alternating temperature multilayered method
on the BaM thick films made by vacuum LPE.
3.1 External Annealed BaM Thin Films Made by Sputtering
We have used target #1 with stiochemetric ratio of B a i^ F e ^ O ig to deposit a
0.15pm BaM thin film on (100) silicon substrate. The sputtering power was 50W and the
final pressure was about
8
x 10' 3 torr. During deposition, the substrate’s temperature was
kept at 620°C. Although we used substrate heating, the film was amorphous in the asdeposited state. After the deposition, the film was nonmagnetic, and insulating. To
crystallize the film, it was heated in air in furnace at 837°C for 10 minutes.
To study the magnetic properties of the 0.15pm BaM thin film, VSM has been used
to measure the hysteresis loops in both perpendicular and parallel directions of the
applied field at room temeprature as shown in figure 3.1. The film was cut into 1.0 cm x
0.6 cm prior to VSM measurements. The magnetization was calculated by dividing the
magnetic moment p by the volume of the film. The average saturation magnetization was
low and about 1000 Gauss which is only about 21% of the BaM bulk value [1]. Some
points were higher than the Ms in figure 3.1 probably due to the substrate effect. The
signal due to the substrate needs to be subtracted from the magnetization data. The
perpendicular squareness was 0.93 and the in-plane one was 0.40. Perpendicular
coercivity of about 4660 Oe and in-plane of about 22000e were obtained. This indicates
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38
a moderate c-axis orientation of the film. To have an excellent c-axis oriented film, the
perpendicular coercivity must be much larger than the in-plane one, the normal
or perpendicular squareness is close to 1 .0 , and the in-plane squareness is as small as
possible. This means that we have all the magnetic domains grown in perpendicular
direction (c-axis is out of the film plane), and very little or no domains grown in the in­
plane direction.
100
d
d
3
E
<D
-50
-100
-15
-10
5
0
5
10
15
H (KOe)
Figure 3.1: The hysteresis loop for the external annealed 0.15pm thin film on Si (100)
using target # 1. The square symbols correspond to an out-of-plane VSM measurement,
and the solid line to an in-plane VSM measurement.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
39
The film texture was detected by x-ray diffraction (XRD) measurements as shown
in figure 3.2. The strongest peaks in this 0.15 pm BaM thin film pattern were from (006)
and (008), which indicates perpendicular c-axis orientation. Both (107) and (114) which
are the strongest peaks in the standard x-ray powder diffraction file were observed. We
also observed the ( 1 1 0 ) peak which corresponds to the in-plane c-axis orientation and to
the fairly wide in-plane hysteresis loop.
40 006
008
107
114
£> 20
O
se
8
v
*»
i r "i r n
20
i i i | i i i i i "r~r i i | i i i i i i i i i | i i i i | i i i i |
30
40
40
2 0 (degree
50
60
Figure 3.2: XRD of the external annealed 0.15 pm BaM thin film on Si (100).
It has been pointed out that the perpendicular magnetic recording layers with
relatively high perpendicular coercivity Hc± (~3 kOe) and large perpendicular squareness
S-Lexhibit higher signal -to-noise (S/N) ratio in perpendicular recording system. Since
our film has high perpendicular squareness, the perpendicular coercivity needs to be
decreased to about 30000e, so they become suitable in perpendicular media.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
40
To check the stiochemetric ratio of this film, EDX was taken for the film as shown
in figure 3.3. Two points were taken from the film. Point 1 was taken from the center of
the film, and point 2 was taken from the edge of the film. The stiochemetric ratio of the
film was Bao.98 Fei 2 0 i9 from point 1 and Bao.88 Fei 2 0 i 9 from point 2 which indicates that
the whole film stiochemetric is almost uniform.
si
loono -
6000--
-'tv—
Acc.Voltage: 20.0 kV Take Off Angle: 12.9 deg
Element N et
Line
Counts
CK
218
OK
2574
A IK
262
S iK
122803
F eK
3957
F eL
1924
B aL
795
B aM
0
Total
Net
Error
+Z-25
+/-S9
+/-71
+Z-448
+/-142
+/-80
+Z-73
+Z-46
k-ratio
(calc.)
Element
Wt.%
...
—
0.459
0.011
57.05
3.08
...
—
0.438
33.22
...
—
0.093
6.66
...
...
100.00
W t%
Error
_
+ /-1.97
+/-0.83
—
+/-1.19
—
+/-0.61
—
Atom % Atom %
Error
—
—
82.43
2.64
+f- 2.85
+ /-0.72
—
...
13.76
+/- 0.49
—
...
1.12
+ /-0.10
...
...
100.00
Figure 3.3: The EDX (taken from the point 1) of the 0.15 pm BaM thin film.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
41
To understand the cause of the in-plane hysteresis, the film microstructure was
studied by SEM. A SEM picture of the 0.15 pm BaM thin film is shown in figure 3.4.
Figure 3.4: SEM observation of the 0.15 pm BaM thin film.
This micrograph shows that the grains consist of two different structures: those with
platelets shape and those which are acicular. It was considered that acicular grains have
their c-axis in the film plane, while platelets grains have their c-axis perpendicular to the
film plane [24]. From the picture, we can see that the platelets grains occupy a larger area
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
42
than the acicular grains. We also see that the acicular grains grows randomly on the
surface of the film, and they grow above the platelets grains.
We have grown another 0.15 pm BaM thin film using target # 2 (the excess barium
target) with stiochemetric ratio of Ba2.22Fei20i9. We used the same deposition conditions
as in the film made by target # 1. In figure 3.5 below, VSM has been used to study the
magnetic properties of the 0.15 pm BaM thin film in perpendicular and in-plane
directions of the applied field.
200
100
-100
-200
-10
■5
0
5
10
H (KOe)
Figure 3.5: The hysteresis loop for the external annealed 0.15 pm thin film on Si (100)
using target # 2. The square symbols correspond to an out-of-plane VSM measurement,
and the solid line to an in-plane VSM measurement.
In comparing with magnetic parameters of the same film made from target # 1, the
perpendicular coercivity decreased to about 3730 Oe and the in-plane one was about
2900 Oe. The perpendicular squareness was also decreased to 0.79 and the in-plane one
decreased to 0.37. However, the average saturation magnetization Ms was about 1800
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
43
Gauss which is about 38% of the BaM bulk value, and it was higher than the average
saturation magnetization M s of the film made from target # 1. It seems that from the VSM
measurements, the c-axis orientation of the BaM film made from target #1 is slightly
better than the one made from target # 2 ( the excess barium target), however, the Ms
value seems to increase with increasing barium ratio in the BaFe^Oig formula. The Ms
value is low even with the BaM film formula is almost the right stiochemetric ratio
BaFei20i9. The EDS data taken for this film (data are not shown) indicate that the atomic
ratio of this film has excess barium percentage; the average formula calculated from three
points taken from the surface of the film (two points from the center and one from the
edge) was about Bai.2Fei20i9. From this we think that the Ms value is not
stiochemtrically related quantity. It must be related to structural issues in the crystal.
The XRD pattern is shown in figure 3.6. Almost all the c-axis peaks are present;
there are small peaks from (107), (114), and (109).
git
S3
.m
40
50
20 (degree)
Figure 3.6: XRD of the external annealed 0.15 pm BaM thin film made from target # 2
on Si (100).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
44
To study the microstructure of the BaM film, SEM was used to explore the surface
of this thin film as shown, in figure 3.7 below. A SEM picture showing the surface is
totally covered with the acicular grains, which have their c-axis randomly oriented and
we believe these grains are responsible for the in-plane coercivity in the VSM
measurement.
Mag = 120.00 KX I— I
Figure 3.7: SEM observation of the 0.15 pm BaM thin film made from target # 2.
To study the thickness effect on the c-axis orientation, we have grown many films
up to 0.9 pm BaM. The substrate (Si (100)) temperature was kept at about 550°C, and the
sputtering pressure of the chamber was about 8 x 10'3 torr. All films were grown from
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
45
target # 2. In figure 3.8 (a) and 3.8 (b) below, we show the graphs of the coercivity and
squareness vs. film thickness.
4500
4000
3500
8
£
I
I
Ts = 550C
3000
- He P er
- He P a ra
2500
2000
1500
1000
500
0
0.2
0.4
0.6
0.8
Film T h ic k n e s s (pm)
Figure 3.8 (a): Coercivity vs. Thickness dependence of the BaM external annealed films.
1
0.9
0.8
0.7
0.6
0.5
-S Per
-S Para
0.4
0.3
0.2
0.1
0
0.2
0.4
0.6
0.8
Film T h ic k n e ss (pm)
Figure 3.8 (b): Squareness vs. Thickness dependence of the BaM external annealed
films.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
46
From figure 3.8 (a) and (b), we can see that the c-axis orientation perpendicular to
film plane is continuing to deteriorate as the thickness increases, which can be seen from
the increase of the parallel or in-plane squareness and decrease of the perpendicular
squareness. Figure 3.8 (a) shows that the films have a random c-axis orientation since the
in-plane coericivity value is close to the perpendicular one. Even with the optimized
sputtering conditions, there is a gradual deterioration of the perpendicular c-axis
orientation as the thickness of BaM films increases. The reason for the deterioration of
the c-axis orientation with the thickness is attributed to the increase of random nucleation
sites over perpendicular nucleation sites [25].
Many efforts have been employed to improve the c-axis orientation by using seed
layers of different kinds such as ZnO [12], andPt [25], or by using different doping
elements substituted for the Ba or the Fe atoms in the BaFei2 0 i9 formula, however the caxis orientation does not improve much except in the Pt underlayer case. However, Pt
underlayer is not suitable for our circulator application because it is a metal.
3.2 In-Situ BaM Thin Films Made by Sputtering
We have used target #2 with stiochemetric ratio of Ba 2 .22 Fei 2 0 i9 to deposit a
0.15pm BaM thin film on (100) silicon substrate. The sputtering power was 50W and the
final pressure was about
8
x 10' 3 torr. During deposition, the substrate’s temperature was
kept at 550°C. Since the films were amorphous after the as-grown state, heating was
required to crystallize them. We have developed the in-situ annealing in which the films
were annealed inside the vacuum chamber. We have introduced 140 torr of argon (80%)
and oxygen (20%) into the chamber. A stainless steel strip heater was used to anneal the
substrates. Current between 55 - 63 amps was used to increase the substrate temperature
up to 850 - 900°C for 10 minutes. The magnetic properties of the in-situ annealed film
were investigated. Both hysteresis loops of the perpendicular and in-plane directions were
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
47
measured by VSM at room temperature as shown in figure 3.9. Prior to VSM
measurements, the film was cut into 1.0 cm x 0.6 cm. The magnetization was calculated
by dividing the magnetic moment p by the volume of the film. The average saturation
magnetization M s was about 2550 Gauss, which is about 54% of the barium ferrite
saturation magnetization bulk value. It seems that the in-situ technique that we developed
is an effective way in improving the average saturation magnetization. The average
saturation magnetization Ms obtained by this in-situ method was about 1.5 times larger
than the one obtained by the external annealing method. Furthermore, when we grew 1.0
pm BaM film using three layers of 0.33 pm thickness each and performing in-situ
annealing process after each layer, the perpendicular Ms increased up to 3950 Gauss
which is about 82% of the bulk value. However, that was at the expense of the c-axis
orientation, which was in random direction with in-plane squareness values higher than
that of the perpendicular one.
250
150
50
-50
-150
-250
-10
■5
0
5
10
H (KOc)
Figure 3.9: The hysteresis loop of the in-situ annealed 0.15pm thin film on Si (100)
using target # 2. The square symbols correspond to an out-of-plane VSM measurement,
and the solid line to an in-plane VSM measurement.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
48
Based on our results, it is seems hard to grow very good c-axis oriented BaM thin
films on Si (100) with high Ms value using the RF sputtering. The in-situ annealed film in
figure 3.9 has perpendicular coercivity of about 3600 Oe, while the in-plane one has
about 3150 Oe, which is close to the perpendicular coercivity. The perpendicular
squareness is about 0.8 and the in-plane one is 0.4. To investigate the c-axis orientation of
the film, XRD was taken. In figure 3.10 we see the c-axis peaks (00/) are present but still
hard to distinguish (0010) and (0012) peaks. Both (107) and (114) peaks are also present.
This means that our film’s c-axis is in random orientation with small tendency toward the
out of plane or perpendicular direction of the film.
JG$
V if5 ;
s
■4mi
0
I
30
- I T T F T1 T
40
50
20 (degree)
Figure 3.10: XRD of the in-situ annealed 0.15 pm BaM thin film on Si (100).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
49
MOKE system was used to check the surface magnetic properties of the in-situ
annealed film; the signal was taken from the center of the film as shown in figure 3.11.
To check the consistency of the data, the hysteresis loop was measured at different points
from the film surface. All data were consistent and almost similar to the hystereysis loop
measured by VSM of the in-situ film in figure 3.10.
1.2
3
n
>.
w
c
0>
£ 0.8
0.6
-10
■5
0
5
10
H (KOe)
Figure 3.11: The perpendicular MOKE hysteresis loop of the in-situ annealed 0.15 pm
BaM thin film on Si (100).
EDX was taken for the film in figure 3.12. Three points were taken from the film
(two points around the center and one at the edge). The average stiochemetric ratio of the
film was calculated from the three points EDX data and was about Bao^FenOig.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50
Si
4000-
1000-
OFe
Ba
Fe
Ba
10
keV
Acc.Voltage: 20.0 kV Take Off Angle: 12.9deg
E lem ent N e t
Counts
Line
CK
42
OK
868
45305
S iK
682
F eL
Fe K
1246
B aM
0
223
B aL
Total
N et
Error
+ /-15
+/-51
+Z-244
+ /-48
H-/-83
+/-25
+ /-43
k- ratio
(c d c .)
Element
Wt.%
—
0.485
59.39
W t%
Error
—
+ /- 3 .52
Atom % Atom %
Error
...
—
84.86
+/- 4.99
_
_
_
_
...
—
...
---
...
---
0.433
34.49
14.12
+/- 0 .94
...
—
0.082
6.12
100.00
+ /-2 .28
—
+ f-l .19
...
—
1.02
+ /- 0.20
100.00
Figure 3.12: The EDX (taken at the center) of the 0.15 pm in-situ annealed BaM film.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
51
The microstructure of the BaM film surface is shown in figure 3.13 below. A SEM
picture showing the surface is totally covered with the acicular grains, which have their caxis randomly oriented. The surface of the film made by the in-situ technique is similar to
the external annealed thin film surface microstructure shown in figure 3.7. In figure 3.13,
the average length of the acicular grains is between 200-300nm. We have studied the
thickness effect using the in-situ technique on the magnetic properties of the films (data
not shown). The c- axis orientation continues to deteriorate as the thickness increases
which is the same result as in the external annealing case (figure 3.8 (a) and (b)).
H ag = '12B.17;KX
|
■'
— I
Figure 3.13: SEM observation of the in-situ annealed 0.15 pm BaM thin film on Si(100).
3Reducing the grain size below lOOnm in order to reduce noise media is very desirable in the ultrahigh
density overcoat free perpendicular magnetic recording [26].
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52
3.3 BaM Films Made by Sputtering on MgO (111) Substrate
To grow the BaM ferrite thick film for the microwave applications, a good c-axis
BaM thin film is used as seed layer for the vacuum LPE technique. We have learned that
it is difficult to grow good c-axis oriented BaM thin film on Si (100). Therefore, we have
grown BaM thin films on MgO (111) substrate. Here the MgO crystalline lattice has a
face centered cubic structure. The lattice parameter of MgO is ciMgo = 4.213A0, such that
a lattice mismatch of e = (asaM - «Mgo)/ «BaM= -0 . 0 1 is expected between atoms on the
MgO (111) plane and those of (00/) BaFei 2 0 i 9
( a BaM
= 5.893A0) [9].
The measured thermal expansion coefficient of MgO (13.6 x 10"06 °C"' from 0 to
1000°C) is greater than that of BaFei2 O i 9 , such that the barium hexaferrite film will
experience biaxial compressive stress upon cooling. Now the stress relationship for the
BaFe^Oig/MgO system indicates that the ultimate obtainable film thickness will be
governed by the fracture strength of the MgO substrate instead of the BaFei 2 0 i 9 film.
W e have grown 0.15 pm BaM thin film on MgO (111) by RF sputtering; the growth
procedure was the same as the one we used in growing BaM thin films on Si (100) except
that MgO (111) substrate was heated between 920°C - 960°C during deposition. The
films after deposition were nonmagnetic; external annealing in air was used to crystallize
the fdms at 840°C for 10 minutes. The BaM thin film was cut into 1.0cm x 0.7cm prior to
VSM measurements. Figure 3.14 below shows the perpendicular and in-plane hysteresis
loops measured by VSM. The perpendicular coercivity was about 1460Oe and 10 times
larger than the in-plane coercivity, however, the perpendicular squareness S-l was low
and only 0.45 while the in-plane squareness was 0.12. The c-axis orientation of the 0.15
pm BaM thin film on MgO (111) has been improved when comparing to the c-axis
orientation of the films grown on Si (100). This is due to the excellent lattice match
between the barium ferrite film and MgO (111) substrate. The average saturation
magnetization Ms was about 2500 Gauss, which is again about 53% of the BaM bulk
value.
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53
200
100
0
-100
-200
-15
-10
■5
0
5
10
15
H (KOe)
Figure 3.14: The hysteresis loop of the 0.15pm thin film on MgO (111). The square
symbols correspond to an out-of-plane VSM measurement, and the solid line to an in­
plane VSM measurement.
The crystal structure of the film is shown in figure 3.15. XRD pattern show good caxis orientation. MgO (222) substrate peak was present, the strongest peaks in the
standard x-ray powder diffraction file (107 and 114) of BaM were quenched. However,
we have the (218) and an unidentified peak at 28°. The c-axis orientation major peaks of
(006), (008), and (0014) were observed.
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54
m
0
0*i
jilj
s 1500
■ '*■.
20
30
50
60
20 (degree)
Figure 3.15: XRD of the 0.15 pm BaM thin film on MgO (111).
The film microstructure was studied by using SEM. Figure 3.16 shows the surface
picture of the 0.15 pm BaM thin film on MgO (111). The surface looks different than the
film deposited on Si (100); no acicular grains were found on the surface. Furthermore,
the surface seems to begin crystallizing but still has no clear morphology.
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55
Mag = 80.00 KX
I
1
Figure 3.16: SEM observation of the 0.15 pm BaM film on MgO (111).
3.4 BaM Films Made by Alternating Temperature
Multilayered Technique on A120 3 (0001) Substrate
We have used AI2 O 3 (0001) or sapphire as substrate to grow BaM thin films.
Sapphire has a rhombohedral crystal structure with a = 5.128 A and a= 55°, whereas
BaM has a hexagonal crystal structure with a = 5.88 A and c = 23.2 A. To match these
lattices, the BaM hexagon has to be sheared. This can be done in one of the two ways
shown in Figures 3.17(b) and 3.17(c). The area changes (AA) of the hexagon after
deformation to the configurations shown in Figures 3.17(b) and 3.17(c) are -25.7 and 09
3.58 A , respectively. This suggests that at the film-substrate interface, the hexagon will
prefer deformations of the form shown in Figures 3.17(c). Thus, it appears that in these
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56
films there are both shear and compressional strains, which take their largest values at the
interface [8 ].
Figure 3.17: Schematic diagram showing the matching of rhombohedral and hexagonal
lattices, (a) Hexagon before the deformation, and (b) one way o f deforming the hexagon
in order to fit it on the rhombohedral lattice. The area change for this case is -25.7 A2, (c)
Another way of fitting the hexagon on the rhombohedral lattice. This configuration gives
an area change of -3.58 A2.
We have developed the alternating temperature multilayered technique to grow 0.3
pm BaM film on AI2 O 3 (0001) using a commercial barium ferrite target. We have used
two BaM layers in which the first was grown at 920 - 960°C, while the second layer was
grown at 800 - 820°C. Also we have used three and five layers to grow 0.45 pm and 0.70
pm BaM film, respectively. All BaM layers utilizing the alternating temperature
multilayer technique were deposited on AI2O3 (0001) single crystal substrate without the
deposition of any seed layer, which simplifies the growth process when compared with
other BaM films grown on AI2O3 (0001) using seed layer. Furthermore, all films in the as
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57
grown state were red brown in color and magnetic, which means that the films were insitu annealed while deposition. Therefore no further annealing was required to crystallize
the films. In this section, we will show the magnetic and structural characterization of the
films grown by this method, and the effect of this method on the c-axis orientation of the
BaM thin films.
Figure 3.18 shows the magnetic properties of the 0.3 pm BaM film on AI2 O 3
(0001). From the hysteresis loops of the thin film, we can see that in-plane loop is almost
linear and closed with small curvature around the origin, indicating that the easy axis is
perpendicular to the film plane. These results about the BaM c-axis orientation are much
better than the ones grown on Si (100) and MgO (111). The perpendicular coercivity in
figure 3.18 was about 1820 Oe and that is about 26 times larger than the in-plane
coercivity. Perpendicular squareness was about 0.76 and the in-plane one was only 0.03
and almost linear, which is very close to the paramagnetic behavior. The saturation
magnetization M s was about 1800 Gauss which is only 38% of the BaM bulk value. The
approximate anisotropy field Ha was calculated from figure 3.18 by extending the
perpendicular saturation magnetization curve and the in plane hysteresis line. Then, the
intersection of these two curves gives the value of H a , which was about 17KOe that is
close to the bulk value of BaM anisotropy field Ha = 17.6 K Oe [27].
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58
200
100
-100
-200
-15
-10
-5
0
5
10
15
H (KOe)
Figure 3.18: The hysteresis loop of the 0.30pm thin film on AI2 O 3 (0001). The square
symbols correspond to an out-of-plane VSM measurement, and the solid line to an in­
plane VSM measurement.
The thin film magnetic parameters in figure 3.18 are consistent with the
mathematical micro-magnetic model4. In figure 3.19, micro-magnetic simulation has
been used to simulate the hysteresis loops for both perpendicular and in-plane loops. The
layer representation of both layers A and B is shown in figure 3.20.
The simulation process starts with entering parameters (such as geometry,
anisotropy constants, exchange coefficients, stochastic field, etc.) in an input file. Then
4
In collaboration with Prof. Richard B. W ells and Dr. Feng Xie.
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59
the program reads those parameters and solves the stochastic Landau-Lifshitz-Gilbert
equation LLG with Ordinary Differential Equation ODE or Stochastic ODE algorithm
[28]. More information about the micro-magnetic simulation is shown in the appendices.
H=16390e
c
=23 Oe
MsA=258emu/cc, MsB=60emu/cd
A=1.3546e-011ergs/cm
A -1 3423e-006ergs/cm
A „=-1.3423e-007ergs/cm
K1A=2.685e+006ergs/cc
K•b=-3.003e+005ergs/cc
8x8x16, «=0.Q5
a=1e-005cm, 5/a=0.0058
tiA=8000 ;hB=7000
-50
— Longitudinal (measured)
Longitudinal (simulated) :
Perpendicular (measured)
Perpendicular (simulated)
-
0.5
0
0.5
Applied field (Oe)
1.5
2
x 10
Figure 3.19: The Simulated Hysteresis loops for the 0.30pm thin film on AI2O3 (0001)
[28],
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60
R*
9 2a
• 2b
• 4ft
• 4f2
• 12k
0 Oxygen
• Ba
Figure 3.20: Layers of A represent layer R and R*, layers of B represent layer S and S*
[28],
In the simulation (figure 3.19), c-axes are within 5° from the perpendicular
direction, which means c-axes are well oriented out of the film plane. The little slope
change in the middle of measured longitudinal loop means the exchange interaction
between layer A and layer B is not strong enough to make them rotate coherently. The
magnetization range of this change is not big, which means MsB is small or cell B is a
small part of the sample. In the simulation, it is assumed that there is 50% A and 50% B
so that M
sb «
M sA . The exchange field is inversely proportional to square of the distance
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61
between cells. To get same order of exchange fields, cells close to each other need
smaller exchange coefficient than that between cells far from each other. Therefore, the
exchange coefficient in the z direction is much less than that in film plane because the
cell to cell distance in the z direction is much less than that in film plane. As expected,
the anisotropy constant of cell A is much greater than that of cell B due to the bi-pyramid
structure in the A layer. However, K ia is less than the anisotropy constant in bulk films.
Simulated coercivity is close to the measured value. Simulated perpendicular Mr is less
than the measured one. Due to the computation capability, this simulated sample is much
smaller than the real film, which means the domain wall movement cannot be simulated
well. This may explain that in the perpendicular direction the loop is not fitted well
before the film saturated in the reverse direction. To improve the magnetic film property,
it is suggested enhancing the exchange interaction between layer A and layer B.
The film structure was investigated by measuring XRD pattern shown in figure
3.21. XRD pattern shows excellent c-axis orientation perpendicular to the film plane.
Almost all the c-axis major peaks were observed with strong intensities. The strongest
peaks in the BaM pattern peak (107) and (114) were quenched, and the (109) peak was
present. These data were consistent with the VSM hysteresis loops in figure 3.18.
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62
1000
20 (degree)
Figure 3.21: XRD of the 0.30 (am BaM thin film on AI2 O 3 (0001).
The film microstructure was studied by using SEM and EDX. Figure 3.22 shows
the surface picture of the 0.30 pm BaM thin film on AI2 O 3 (0001). The surface looks
different than the film deposited on Si (100). No acicular grains were found; the surface
is sponge like shape and probably spreading all over the surface.
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63
Mag = 32.42 KX
^
Figure 3.22: SEM observation of the 0.30 pm BaM film on AI2 O 3 (0001).
In figure 3.23, we show the surface image of the 0.30 pm BaM film on AI2O 3
(0001). Six points were taken on the surface to make the EDX scan for checking the
stiochemetric ratio of this film. In figure 3.24, EDX scan was taken for the whole area
(taken as one EDX scan) of the surface and was compared to the scan taken from the six
points. The average stiochemetric ratio of these six points is about Bao.65 Fei2O i 9 , and the
stiochemetric ratio of the whole area is about Bao.75Fe12Oi9, averaging these two formulas
we have about Bao.7 oFei2 O i 9 , which is deficient in barium. One possible cause of the
barium deficient film is the scattering loss mainly caused by the bombardment of highenergetic ions from the plasma [13, 29].
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64
Figure 3.23:
6
points on the surface of the 0.30 pm BaM film on AI2 O 3 (0001).
The barium deficient surface of the 0.30 pm BaM film makes us believe that we
have excess iron oxide of some phase in existence at the surface. However, the XRD
pattern of this film did not show any iron oxide (of any phase) relevant peaks. This could
be due to the fact that our BaM c-axis peaks are so strong, well diffracted, and dominant
that any iron oxide peak is indiscernible.
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1400
1200
1000
800
600
400
200
0
fceV
Ace.Voltage: 20.0 W Take Off Angle; 12.9 deg
Element Net
Counts
Line
CK
817
OK
8112
AIK
10326
FeK
4669
3282
FeL
BaL
689
BaM
0
Total
Net
Etror
+/-5S
+/-146
+/-I27
+/-132
+7-143
+/-106
+Z-33
k-iatio
(cald)
Element wt.%; Atom% Atom%
Wt.% Etror
Etror
—
—
—
—
—
0.588
0.1®
0.210
63.64
23.35
11.26
+/-1.14
+/4L25
+/4K32
78.65
17.11
3.99
+/-1.42
+/- 021
+/-0.I1
—
—
—
—
0.033
—
1.74
—
100.00
+/-027 : 025
—
—
—
+/- 0.04
—
100.00
Figure 3.24: The EDX data taken for the surface image of figure 3.23.
In figure 3.25 (a) and (b) below, we show the mapping of selected area (about
1.5mmxl.5mm) of the 0.30 pm BaM film surface.
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66
Fe K
$slf
p llffe # W # li
(a)
(b)
Figure 3.25 (a): Barium EDX mapping and (b): Fe EDX mapping of a selected area of
the 0.30 pm BaM film surface.
Qualitatively speaking, we can see that over a selected area (about 1.5mmx 1.5mm)
of the 0.30 pm BaM film surface, the individual pixels of barium and iron are evenly
distributed over that area of the surface.
It seems that the low barium ratio in the molecular formula of BaFenOig is the
reason for the low Ms value of the film. We think that the surface has excess iron oxide
which is probably belong to the a-Fe 2 C>3 phase. This phase exists below 910°C, which is
the case when we deposited the second layer at 800 - 820°C.
To study the relation between thickness and the magnetic parameters of the thin
films, we have grown 0.45 pm and 0.70 pm BaM film using three and five layers. All
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67
BaM layers utilizing the alternating temperature multilayer technique were deposited on
AI2 O 3 (0001) sapphire substrate. The VSM hysteresis loops in perpendicular and in-plane
directions of the 0.45 pm are shown in figure 3.26. The c-axis orientation is still well
oriented perpendicular to the film plane; however it is a little bit deteriorated when
compared to the 0.30 pm BaM thin film. In figure 3.26 the perpendicular coercivity was
2050 Oe and 5.5 times larger than the in-plane one which was about 370 Oe, the
perpendicular squareness was 0.87 and only 0.07 squareness was measured in the in­
plane direction, average saturation magnetization is still low though and only 1400
Gauss.
150
100
50
0
-50
-100
-150
-15
-10
-5
0
10
15
H(KOe)
Figure 3.26: The hysteresis loop of the 0.45pm thin film on AI2O3 (0001). The square
symbols correspond to an out-of-plane VSM measurement, and the solid line to an in­
plane VSM measurement.
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68
Hematite or a-Fe 2 0 3 phase is antiferromagnetic material [1], as discussed in section
1.5. The antiferromagnetic material has no net magnetization, therefore, the low Ms value
of the thin film deposited on AI2 O 3 (0001) using the alternating temperature multilayered
technique is probably due to this antiferromagnetic phase. It seems that the low barium
percentage in the molecular formula of the film has small effect on the c-axis orientation.
We think that the effect of a-Fe 2 C>3 phase on the c-axis orientation would be by partially
pinning the well c-axis oriented BaM thin film of the layer beneath it. Therefore, when
another BaM layer (third layer) is deposited at 920 - 960°C above the barium deficient
layer (the second layer), the a-Fe 2 0 3 phase will be sandwiched between two well c-axis
oriented BaM films, and this sandwiched layer will serve as pinning layer to the BaM
layer beneath it. This will prevent a direct exchange interaction between the first and the
third BaM layers, therefore, boosting the c-axis orientation when the film thickness
increases. We think that pinning BaM layer depends on the degree of formation of aFe 2 0 3 phase during the deposition of the second layer.
The XRD pattern of the 0.45 pm thin film is shown in figure 3.27. The crystal
structure of the 0.45 pm thin film is the same as the one in figure 3.21 of the 0.30 pm thin
film. It can be seen clearly from figure 3.27 that the c-axis orientation in perpendicular
direction to the film plane is the dominant phase since all the c-axis peaks are present.
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69
00
Q
8001
600
*■
s
«.
Cl
400
H
■
a 200
20 (degree)
Figure 3.27: XRD of the 0.45 [am BaM thin film on AI2 O 3 (0001).
The hysteresis loops of the 0.7 pm BaM thin film and its XRD pattern are shown in
figures 3.28 and 3.29 respectively.
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70
150
100
50
-50
-100
-150
-15
-10
■5
0
5
10
15
H (K O e)
Figure 3.28: The hysteresis loop of the 0.7pm thin film on AI2 O 3 (0001). The square
symbols correspond to an out-of-plane VSM measurement, and the solid line to an in­
plane VSM measurement.
008
006
a
w
109
0014
ssifei
20
30
40
50
20 (cleg)
Figure 3.29: XRD of the 0.70 pm BaM thin film on AI2 O 3 (0001).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
60
71
The perpendicular coercivity of the 0.70 pm BaM thin film was about 2330 Oe and
perpendicular squareness of 0.77. We have noticed an increase in the in-plane coercivity
to about 700 Oe and in-plane squareness of about 0.12. This means that the c-axis
orientation of the 0.70 pm BaM thin film is deteriorated a little bit when compared to the
c-axis orientation of the 0.3 pm and 0.45 pm BaM films. However, the c-axis orientation
of the 0.7 pm film is better than the c-axis orientation of the BaM thin films grown on Si
(100) and M g O ( lll) .
The XRD pattern in figure 3.29 above shows that we have peaks belongs to the aFe 20 3 phase. The two strongest peaks in the a-Fe 2 0 3 phase (104) and (012) are observed
accompanied with the in-plane a-Fe 20 3 phase peak (110). These peaks were not observed
in the 0.30 pm BaM thin film XRD pattern (figure 3.21), because they were having very
small intensities compared to the strong intensities of the BaM c-axis oriented peaks.
When the c-axis orientation deteriorated a little bit, their intensity decreased and that
allowed the a-Fe 2 C>3 phase to be visible. Also from the XRD pattern in figure 3.29 we see
the c-axis peaks are present with some other peaks like (203) and (109).
To compare the c-axis orientation of the 0.70 pm BaM thin film using the
alternating temperature multilayered technique with the one deposition layer technique,
we have grown a 0.7 pm BaM thin film on AI2 O 3 (0001) at the same conditions of the
film grown by temperature multilayered technique. In figure 3.30 below, we show the
hysteresis loop of the 0.7 pm BaM thin film on Al 2 0 3 (0001) using one deposition layer.
The substrate temperature was kept at 920 - 960°C while depositing the film. The
perpendicular coercivity has decreased to 1200 Oe, which is about 1.38 times larger than
the in-plane coercivity. This ratio is lower than that in the film grown by the alternating
temperature multilayered technique. The average saturation magnetization Ms of the one
layer deposition technique was about 2400 Gauss, which is higher than that of the film
grown by the alternating temperature multilayered technique.
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72
200.0
100.0
c
■sH
S
V
0.0
V
-
100.0
-
200.0
-15.0
-
10.0
-5.0
0.0
5.0
10.0
15.0
H(KOe)
Figure 3.30: The hysteresis loop of the 0.7pm thin film on AI2 O 3 (0001) using one
deposition layer. The square symbols correspond to an out-of-plane VSM measurement,
and the solid line to an in-plane VSM measurement.
The SEM surface picture of the 0.7 pm BaM thin film on AI2O3 (0001) is shown in
figure 3.31. We believe that the surface is containing of sponge-like domains, and looks
like the surface of the 0.30 pm BaM film shown in figure 3.22. However, this 0.7 pm
BaM film shows some regions of the sponge-like domains start to nucleate and crystallize
to possibly a barium hexaferrite platelet where their c-axis is oriented perpendicular to
film plane. Also we can see that there are many outgrowth platelets (with average size
was about 1 pm) have their c-axis oriented parallel to the film surface. We believe that
these outgrowth platelets are responsible for increasing the coercivity in the in-plane
direction, and therefore deteriorating the c-axis orientation.
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73
M a g = 4 2 .3 3 K X
^
Figure 3.31: SEM observation of the 0.70 pm BaM film on AI2 O 3 (0001).
In figure 3.32 below, we show the image of the 0.70 pm BaM film surface. Four
points were taken on the surface to make the EDX scan for checking the stiochemetric
ratio of this film. In figure 3.33, EDX scan was taken for the whole area (taken as one
EDX scan) of the surface and was compared to the scan taken from the four points. The
average stiochemetric ratio of these four points is about Bao.99 Fei 2 0 i 9 , and the
stiochemetric ratio of the whole area is about Bao.94 Fei 2 0 i9 , averaging these two formulas
we have about B a o 96Fei20i9, which is almost the right stiochemetric ratio for BaM
molecular formula.
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Figure 3.32: 4 points on the surface o f the 0.70 pm BaM film on AI2 O3 (0001).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
75
150 0 -
1000-
500-
Fe
Ba
Fe
Da Ba
m
Acc. Voltage: 20.0 kV Take O ff Angle: 12.9 deg
Elem ent N et
Line
Counts
OK
4675
19359
A IK
F eK
2190
F eL
977
B aL
402
B aM
0
Totsi
N et
Error
+ /-90
+/-122
+Z-96
+/-61
+/-45
+Z-25
k-ratio
(calc.)
0.439
0.409
0.128
Elem ent
wt%
Wt.%
Error
+ /-1 .03
+ /-0.25
+/-0.26
53.54
39.33
5.95
. ..
—
0.025
1.18
...
...
100.00
—
+ /-0.13
...
Atom % Atom %
Error
68.03
+ /-1 .3 1
29.63
+ /-0 .1 9
+/- 0.09
2.16
...
—
0.17
+/- 0.02
. ..
...
100.00
Figure 3.33: The EDX data taken for the surface image of figure 3.31.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
76
In figure 3.34 (a) and (b) below, we show the mapping of selected area (about
1.5mmxl.5mm) of the 0.70 pm BaM film surface.
(a)
(b)
Figure 3.34 (a): Barium EDX mapping and (b): Fe EDX mapping of a selected area of
the 0.70 pm BaM film surface.
From the VSM and XRD data we can see that the alternating temperature
multilayered technique is an effective method to grow very good c-axis oriented films up
to 0.7 pm thickness. More research needs to be done to determine the effect of the
thickness on the magnetic behavior of the films beyond 0.7 pm thickness. The average
saturation magnetization of the films prepared by the alternating temperature
multilayered technique is, however, still low because of the formation of a-Fe 2 0 3 phase
which adds no magnetization to the film since it is antiferromagnetic substance. One
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77
possible solution to improve the c-axis orientation as the thickness increase is to grow the
second layer of BaM at lower temperatures (lower than 820°C).
3.5 BaM Thick Films Made by RF Sputtering
In our attempt to grow BaM thick films beyond 1 pm, we have used rf sputtering
system to grow thick films on Si (100). Since our sputtering rate is low and about 0.10.15 pm / hour, we had to sputter for a long time to deposit 14 pm BaM thick film. The
films after deposition were black in color, and covered with red powder which we believe
it belongs to a-Fe 2 C>3 phase. We also noticed that the films sometimes tend to peel off the
substrate. In figure 3.35 we show the hysteresis loop of the 14 pm BaM thick film. The
perpendicular and in-plane coercivities were close to each other, the in-plane squareness
was higher than the perpendicular one. This means that the c- axis orientation was in
random direction.
To build the self biased circulator, thickness of the BaM film should be at least 100
pm and higher. We can see that RF sputtering is not very promising in making very thick
films.
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78
4
1
0
1
1
3
4
-15
-10
•5
0
5
10
15
H(KCte)
Figure 3.35: The hysteresis loop of the 14pm thick film on Si (100). The square symbols
correspond to an out-of-plane VSM measurement, and the solid line to an in-plane VSM
measurement.
3.6 BaM Thick Films Made by vacuum LPE and Reflow
Method
The BaM thick film was grown by the vacuum LPE and the reflow method as
discussed in section 2.5. After one hour reflow of the BaM crystal chunk on top of the
sapphire AI2 O 3 substrate, the average thickness of the film was between 300-550 pm.
This is relatively high growth rate for the BaM thick films, which is cost and time
effective. Furthermore, our BaM thick film was deposited on bare sapphire AI2 O 3
substrate. No seed layer was necessary to grow the BaM thick film. The thickness of the
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79
film was dependent upon the size of the precut solidified specimen and the size of the
substrate. Films with thicknesses of 0.2 to 0.5 mm are routinely obtained.
We have used the VSM to study the magnetic properties of our thick film. In
figure 3.36 the hysteresis loops for both perpendicular and parallel directions of the 350
pm thick film are shown. Most of the previous work on growing BaM thick films by the
LPE produced a single crystal structure films with coercivities in perpendicular direction
of about 10 Oe [22]. Our BaM film has coercivity of about 100 Oe in the perpendicular
direction. The average saturation magnetization value
4 7 tMs of
our film was relatively
low about 1500-2000 Gauss at room temperature [30].
200
150
100
50
0
-50
-100
-150
-200
-15
-10
-5
0
5
10
15
H(KOe)
Figure 3.36: The hysteresis loop of the 350pm thick film on (0001) sapphire AI2O3
substrate. The square symbols correspond to an out-of-plane VSM measurement, and the
solid line to an in-plane VSM measurement.
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80
X-ray diffraction pattern (XRD) in figure 3.37 below shows that the 350 pm thick
film has out-of-plane uniaxial anisotropy where the c axis of the film is aligned normal to
the film plane. Intense peaks identified as the 006, 008 and 0014 reflections of the
magnetoplumbite phase are observed.
008
006
6000 -
5000 -
0014
2000 -
1000 -
0010
0016
20
30
40
50
60
20 (degree)
Figure 3.37: XRD pattern of the 350 pm BaM thick film on (0001) sapphire AI2 O 3
substrate.
Figure 3.38 (a) shows a SEM image of a typical 350 pm thick BaM film grown on a
sapphire (0001) substrate. Here the BaM hexagonal platelets can be clearly seen on the
surface of the film. They are randomly oriented with respect to each other, but they are all
c-axis oriented parallel to the surface normal (out of the plane of the figure). The
thickness of 350 pm was measured by cross sectional SEM and is shown in figure 3.38(b)
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81
The BaM film has a coercivity of approximately 100 Oe in the perpendicular
direction. We speculate that the fairly large size of the BaM crystals (average width of 20
pm) attributes to the relatively low value of the coercivity. A possible way to improve the
coercivity in the future could be by controlling the cooling rate to facilitate the growth of
smaller size crystals which will inherently boost Hc [30].
Figure 3.38 (a): SEM surface image of the 350 pm BaM thick film on sapphire (AI2O3)
(0 0 0 1 ) substrate.
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82
Figure 3.38 (b): SEM cross section image of the 350 pm BaM thick film.
3.7 BaM Thick Films Made by Using BaM Thin Film Seed
Layer
We have used BaM thin films grown by the alternating temperature multilayered
technique as seed layer, then we used the reflow method to make the thick film. Since the
0.30 pm BaM thin film shows excellent c-axis orientation as described in section 3.4, we
employed this thin film as seed layer. A sample piece from a solid melt, which has been
prepared by the vacuum LPE, was placed on the thin film and then heated to 1200°C for 3
hours in air. Then the film was cooled at cooling rates between 3 to 5 degrees Celsius per
minute. VSM hysteresis loop of the thick film is shown in figure 3.39.
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83
u
s
£
-40
-60
-80
-15
-10
■5
0
5
10
15
H(KOe)
Figure 3.39: The hysteresis loop of the 250pm BaM thick film using BaM thin film as
seen layer. The square symbols correspond to an out-of-plane VSM measurement, and
the solid line to an in-plane VSM measurement.
The perpendicular coercivity was about 110 Oe and the in-plane one was 50 Oe.
The squareness was low in both directions. We have tried to grow BaM thick films at
different reflow temperatures using the same thin film seed layer. The hysteresis loop of
the thick film that was grown at 1100°C is similar to the one in figure 3.39. Larger
coercivities were observed in both directions when we grew a BaM thick film at 1150°C
for 3 hours. In figure 3.40 below we show the BaM thick films grown by using seed layer
dependence on the reflowing temperature.
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84
800
700
600
o
©
500
>»
-w
400
"3
U
u 300
o
U
200
100
1000
1050
1100
1150
1200
1250
Temperature C
Figure 3.40: The BaM thick films grown by using seed layer dependence on the
reflowing temperature. The square symbols correspond to an out-of-plane coercivity and
the solid line to in-plane coercivity.
The BaM seed layer seems to deteriorate the c-axis orientation of the thick film as
seen from figure 3.40 at temperatures lower than 1200°C. However, we have similar
results obtained at 1200°C for both BaM thick films grown with or without seed layer. In
order to make a conclusion about the effect of the seed layer on the structural, growth and
magnetic properties of the thick films, more work needs to be done. In figure 3.41(a) and
(b) below, we show a SEM picture of the cross section where hexagonal platelets can be
seen clearly and a surface picture of the thick film grown by vacuum LPE using BaM
seed layer.
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85
Mag = 1.53 KX
20fjm
I
1
Figure 3.41 (a): SEM cross section image of the BaM thick film using seed layer.
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86
Figure 3.41 (b): SEM surface image of the BaM thick film using seed layer.
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87
CHAPTER 4
Conclusions and Future Work
We have grown barium ferrite thin films by RF sputtering on Si (100), MgO (111)
and AI2 O 3 (0001) substrates. The c-axis orientation for the BaM thin films grown on Si
( 1 0 0 ) was not oriented very well in the direction of the easy magnetization direction, i.e.
perpendicular to film plane. W ith the external annealing method, perpendicular
coercivities and squareness of about 47000e and 0.93 respectively were obtained.
In our attempt to improve the c-axis orientation of the films as the thickness
increases, we have developed the in-situ technique to crystallize the films grown on Si
(100). We have increased the average saturation magnetization Ms up to 3950 Gauss
which is about 82% of the bulk value using three thin in-situ annealed BaM layers,
however, the c-axis orientation did not improve and continues to deteriorate as the
thickness of the film increases. It seems it is hard to grow high quality c-axis oriented
BaM thin films on bare Si (100) with high Ms value. BaM thin films grown on MgO
(111) show better c-axis orientation and a moderate M s value, the perpendicular
coercivity was about 10 times larger than the in-plane one. We believe that the excellent
lattice match between MgO (111) and BaM film (about 1% mismatch) has improved the
c-axis orientation in the perpendicular direction.
We have developed the alternating multilayered technique to grow good c-axis
oriented up to 0.7 pm BaM thin films. Using this method, we grew the antiferromagnetic
phase (a-Fe 2 0 a) and used it as a pinning layer to grow BaM thin films on AI2O3 (0001)
without any seed layer. The perpendicular coercivity was about 26 times larger than the
in-plane one. We believe that the M s value was low because of the formation of a-Fe 2 0 3
phase. The c-axis and magnetic parameters are consistent with our mathematical micromagnetic model. XRD measurements verify the hexagonal structure of the film.
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88
Hysteresis loop measurements are consistent with the mathematical model for c-axis
angles within 5° off the perpendicular.
The RF sputtering is not very effective in growing BaM thick films. We have
introduced the vacuum LPE and the reflow method to grow films with thicknesses of 0.2
to 0.5 mm on AI2 O 3 (0001) substrates. The films were having up to lOOOe perpendicular
coercivities which mean that they have polycrystalline structure. Controlling the cooling
rate would possibly facilitate the growth of smaller size crystals which will inherently
boost Hc and help to fabricate the self biased circulator.
BaM thin film were grown on AI2 O 3 (0001) substrates and used as seed layers to
grow the BaM thick film using the vacuum LPE and the reflow method, we have studied
briefly the effect of the seed layer on the magnetic properties of the thick films, based on
our results, it seems that the seed layer did not increase the perpendicular Hc up to high
levels when compared to thick films without seed layer. However, more research needs to
be done to determine whether the BaM thin film seed layer thickness has an effect on the
magnetic behavior of the thick films, also more research is required to study the effect of
the reflow temperature, time of heating, different cooling rates at different temperatures
on the growth and magnetic properties of the films.
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89
Bibliography
[1]
B. D. Cullity, Introduction to magnetic materials. Addison-Wesley Publication
Co, 1972.
[2]
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Ryan S. Adams, Bandwidth Optimization of an Integrated Microstrip Circulator
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Y. Hoshi, Y. Kubota, and H. Ikawa, Journal of applied physics 81(8), Apr. 1997,
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T. L. Hylton, M. A. Parker, M. Ullah, K. R. Coffey, R. Umphress, and J. K.
Howard, Journal of applied physics 75(10), May 1994, 5960
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Y. Hoshi, Y. Kubota, and M. Naoe, IEEE Transactions on Magnetics Vol. Mag31, No. 6 , Nov. 1995, 2782.
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S. G. Wang, S. D. Yoon, and C. Vittoria, Journal of applied physics 92(11), Dec.
2002, 6728.
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S. D. Yoon, and C. Vittoria, and S. A. Oliver, Journal of applied physics 92(11),
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M. S. Yaun, H. L. Glass, and L. R. Adkins Applied physics letters 53(4), Jul.
1988, 340.
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Xiaoyu Sui, and M. H. Kryder, Applied physics letters 63(11), Sept. 1993, 1582.
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J. Rousseau, Journal of applied physics 93(12), Jun. 2003, 9898.
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S. D. Yoon, and C. Vittoria, IEEE Transactions on Magnetics Vol. Mag-39, No.
5, Sept. 2003, 3163.
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S. Foner, Rev. Sci. Instr., 30,1959, 548.
[24]
Xiaoyu Sui, M. H. Kryder, Bunsen Y. Wong, and D. E. Laughlin, IEEE
Transactions on Magnetics Vol. Mag-29, No. 6 , Nov. 1993, 3751.
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Z. Zhuang, M. Rao, D. E. Laughlin, and M. H. Kryder, Journal of applied physics
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Z. Zhuang, M. Rao, R. M. White, D. E. Laughlin, and M. H. Kryder, Journal of
applied physics 87(9), May 2000, 6370.
[27] H. Kojima, Ferromagnetic materials. Vol. 3, north holland publishing co., Newyork, 1982.
[28]
Feng Xie, A Micromagnetic Model of Barium Ferrite For Microwave Circulator
Design, Ph.D. Dissertation, University of Idaho, Jul. 2006.
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A. Morisako, H. Nakanishi, M. Matsumoto, and M. Naoe, Journal of applied
physics 75(10), May 1994, 5969.
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[30]
Y. Kranov, A. A buzir,, T. Prakash, D. N. Mcllroy, and W. J. Yeh, IEEE Transactions
on Magnetics, Vol. Mag-42, No. 10, Oct. 2006, 3338.
[31]
Feng Xie, A Micromagnetic Model of hexaferrite and some simulation results,
master thesis, University of Idaho, 2004.
[32]
W. F. Brown, Jr., Micromagnetics, Wiley-Interscience, New York, 1963.
[33]
Soshin Chikazumi, Physics of ferromagnetism, 2nd Edition, Oxford science
publications, 1997
[34]
T. L. Gilbert, IEEE Transactions on Magnetics, Vol. 40, No. 6 , 2004, 3443.
[35]
Jian-Gang Zhu, Interactive phenomena in magnetic thin films, Ph.D. dissertation,
University of California, San Diego.
[36]
R. Moskowitz and E. D. Torre, IEEE Transactions on Magnetics, Vol. 2, No. 4,
Dec. 1996,739.
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[38]
J. Kanamori, Anisotropy of magnetostriction of ferromagnetic and
antiferromagnetic materials, Academic Press, 1963.
[39]
C. Kittel, Physical review, vol. 70, Dec. 1946, 965.
[40]
C. Kittel, Physical theory of ferromagnetic domains, Review of modem physics,
Vol. 21, No. 4, Oct. 1949, 541.
[41]
W. F. Brown, Jr., Physical review, vol. 130, June 1963, 1677.
[42]
J. L. Garcia-Palacios and F. J. Lazaro, Physical review B, vol. 48, Dec. 1998,
14958.
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92
Appendix A
Landau - Lifshitz - Gilbert - Equation
A.l Geometric Structure
(a) Top view
(b) Side view
Figure A l: Top and side views of geometric arrangement of the hexagonal cells.
In simulating the hysteresis loop in this dissertation, we were concerned with the
modeling of barium ferrite thick films that possess hexagonal structure. The geometric
structure is the same as that in [31]. In our model, a thick film is made up of multi-layer
hexagons. The hexagonal cells are arranged in a hexagonal lattice, as shown in figure A l,
with a lattice constant a. Each hexagon represents a crystallite that is referred to as a cell.
In the calculation hexagons in the film are approximated as being geometrically identical.
The surface-to-surface boundary separation of the adjacent hexagons is denoted as d. For
every cell the neighbor in the top layer is right above it. The distance from layer bottom
to layer bottom is denoted by<5! Height of the gap between two adjacent layers is denoted
by dz. The coordinate is set such that the z = 0 plane is the bottom layer and the z-axis is
pointing up.
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93
A.2 Physical Equations
Micromagnetics [32] is a phenomenological approach pioneered by W. F. Brown, Jr. It
assumes that a sufficiently small volume (called a magnetic cell or cell in this
dissertation) of magnetic material possesses a spontaneous saturation magnetization M s ,
which is dependent on the temperature but independent on the magnetic field. However,
the direction of M s depends on the total magnetic field in the cell so that it is represented
by a magnetization vector M . The rotation of M follows the Landau-Lifshitz (LL)
equation [33]:
— = - y LM x H
^ - \ m x {m x h )1.
dt
4 tM s
(A .l)
Or, equivalently [31] the Landau-Lifshitz-Gilbert (LLG) equation [34]:
dM
a a dM
x ----= - y r M x H +---- M
dt
0
M,
dt
(A.2)
where H is the vector of total effective magnetic field in Oe; f L, f G > 0 are the
gyromagnetic constants in
of magnetization in
H z /O e ;
e m u /c c ;
X is the relaxation frequency in
M s is the saturation magnetization in
;
M
e m u /c c ;
is the vector
and a is a
phenomenological damping constant. In fact, the LL equation and the LLG equation are
equivalent to each other with
a =
X
and yG = y L(\ + a 2).
4 7ryLM s
(A.3)
When a « 1 , y L ~ y G = 2 .S M H z /O e . In consequence, there are several equivalent forms
of the LL equation. In this dissertation, the equation used in the simulations is
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94
A.3 Field Analysis
A.3.1 External Applied Field
The external applied field can be a DC or an AC field. In hysteresis loop
measurements the external applied field is a DC field, which is represented by H app. In
microwave communications, there exists an external AC field.
A.3.2 Demagnetizing Field
The demagnetizing field is caused by equivalent magnetic dipoles in magnetic
materials and on the surfaces. Since magnetization is assumed uniformly distributed in
magnetic cells, there is no volume magnetic charge in cells and the demagnetizing field is
caused by magnetic dipoles on the surfaces of cells:
N
(A.5)
where i is the index of the target cell, 7 is the index of source cells, and
(A.6 )
is called magnetostatic interaction matrix [35]or magnetometric demagnetization tensor
[36]. The magnetostatic interaction matrix Dtj includes two parts: D u caused by
magnetic dipoles on the surfaces of the target cell itself and D i} (i ^ j ) caused by
magnetic dipoles on surfaces of other cells. The Dti satisfies the sum rule [36]:
T r(b u ) = 1 (in SI units) or 7v(d(V) = 4 n (in CGS units).
The magnetostatic interaction matrix D u of a film disk can be expressed as
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(A.7)
95
4
=
Nt
0
0
0
Nt
0
0
0
JVj.
(A.8 )
where N t is the demagnetizing factor in the film plane and N ± is the demagnetizing
factor in the perpendicular direction.
Equations of D u for an ellipsoid are given in [37]. Equations of Dtj on a hexagonal
lattice are given in [31]. In order to reduce the computation time of the demagnetizing
field the Fast Fourier Transform (FFT) is applied [31].
A.3.3 Crystalline Anisotropy Field
Magnetocrystalline anisotropy is caused by spin-orbit coupling and spin-spin
coupling [38]. It is usually presumed to have the same symmetry as the crystal structure
of the material. There are two important principle cases of magnetocrystalline anisotropy:
uniaxial and cubic anisotropy.
Uniaxial anisotropy exists in hexagonal materials. The uniaxial anisotropy energy
( E au) depends on the angle 0 between c-axis ( c ) and the reduced magnetization vector
(W = m / M s ):
E a,u = K lu sin 2 0+ K 2u sin 4 0 ,
(A.9)
where K lu and K lu are anisotropy constants of the material for uniaxial anisotropy.
The corresponding effective magnetic field is
H a,u
(A. 10)
where
(A .ll)
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96
Cubic anisotropy exists in cubic crystals. The anisotropy energy can be expressed in
terms of the direction cosines of the magnetization vector with respect to the three cube
edges. In other words, the cubic anisotropy energy depends on the projection of the
magnetization vector on three axes ( c , , c2, and c3):
E a,c = K u t e •f Y & ■f Y + (c2 • f Y t e • f Y + t e •f Y (q • f Y ].
(A . 1 2 )
where K lc is the anisotropy constant for cubic anisotropy. The corresponding effective
magnetic field is
TJ
dEac
a'c
_
M sdy/
a’c ~
Y
1
=—
2K
T T ^ {(c
M
(A. 13)
•
v l x-
(? 1 •
v YEi +
(? 2 •
v)lx-
(c 2 • ^ )1 ? 2 + (c 3 •
y/)[\ -
(c 3 •
}.
s
A.3.4 Exchange Field
Magnetizations of magnetic materials are mainly due to electron spins. From
quantum theory, the effective coupling between spins is equivalent to a potential energy
of the form [39]:
V ^ -lJ ^ -S j
(A. 14)
where J tj is the exchange integral connecting atoms i and j , and S) (a vector with
magnitude S ) is the spin angular momentum of atom i measured in multiples of hj27i.
In a micromagnetic model the direction of S ( in a cell is represented by the direction of
magnetization (ip). With the consideration of exchange interaction only between nearest
neighbors and J tj being constants J , Kittel [40] derived the expression of the exchange
energy density as
«, = 4 ? ¥ , )2+ (v V, f + (Vyrz Y J
where A is the exchange stiffness constant given by [33],
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(A. 15)
97
J S 2/ a s for a simple cubic lattice
A = 2 J S 2/ as for a body - centered cubic lattice.
(A. 16)
4 J S 2/ a s for a face - centered cubic lattice
where a s is the spin lattice constant. Taking into consideration \j) • y/ = 1, an equivalent
form of ex is [33]
ex = —A ijjr • A\fir.
(A. 17)
Following equation (A. 15), the exchange energy density between two neighbor cells can
be expressed as
e^
A^r.
(a . is )
A ij
where
is the distance between cell i and cell j . As a result, the effective exchange
field is
M ,tiy ,
(A19)
A M si
If one only takes into consideration the exchange energy due to nearest neighbors
(NN), the effective exchange field is
H xi = —
M1 ,i.i
1V
X ~
^ J2
A2
JeNN
■
(A.20)
A y
Taking into account \j/. •\jiri = 1, the exchange energy can also be expressed as
1 - w . xj)
4-4 -2 A2.
y
.
e = 2A
(A.21)
so that the effective exchange field is
H x i=—
M
M
Y
^ . ~ ^AT L -
s,i je N N
(A -22>
A tj
For energy analysis or for solving the LL equation, there is no difference between
equation (A.20) and equation (A.22) since\j/i •ij/i =1 andif/i x y /i = 0 . However, for FMR
analysis, the self-biasing field will be different and equation (A.20) means at the
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98
saturation state the FMR frequency is independent of the exchange field, which is more
reasonable. For this reason, equation (A.20) is applied in simulations.
Since it is hard to measure Ai}, another parameter in simulations is defined
K j= - A -
(A.23)
A i j K lu
which reflects the relative strength of the exchange field compared to the anisotropy
field H Kiu= ^ - .
Ms
A.3.5 Stochastic Field
Thermal fluctuations exist in ferromagnetic materials [41], In this case, the rotation
of M follows the stochastic Landau-Lifshitz-Gilbert equation [42]:
dij/ = {- yLi f j / x H ) - a y L
x x / / -
y ,(i/sx d h )-a y L
x x d h )]
(A.24)
where h is a stochastic field modeled as a Gaussian process with the property
£(£(*)) = 0 , E(ht {t)hj (t)) =
j M sv
S y S it-s)
(A.25)
w hereE( ) is expectation; k B is the Boltzmann’s constant; T is the temperature; v is the
volume; and S is the Dirac S.
Since the integration of dt is proportional to t and that of dh is proportional to y fd t, in
order to compare the stochastic field with the anisotropy field, it defines
2^
T
V TM.vt
so that dh = hs4 d t .
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(A.27)
99
A.3.6 Total Field
In summary, the magnetic fields under consideration are the external applied field,
demagnetizing field, crystalline anisotropy field, exchange field, and stochastic field. The
rotation of M follows equation (A.24), in which
H = H app+ H dem+ H ani+ H exc
and
H„
' exc
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(A.28)
Appendix B Demagnetizing interaction matrix
In appendix A , the demagnetizing interaction matrix is defined as
f
vJ,
The average demagnetizing field over the volume v of the ith grain
ft
iiisxig
jW
'M j
Defining an interaction matrix f as
lr - r f
ii M
then we have
£ * ^ -iT ^ d x d y d z,
Vv,
Djj* = I jT * * d xtfy< k,
V V,
*S
85 “ / T**dxdydz»
£3>^ ^ i j r ^ d x t f y d z ,
Vv,
D * = ~ j T z*dxdydz,
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101
D * ~ - j T a dxdydz,
v Vj
” "fT^dbcdydz,
Dy ~ —\ T a dxdydz ,
w ith
1
. ‘J s
2
,2
&
4 ’
y i= ~ * + i;y >
"M '
«4M
2
A .-x
2
A ,+ x
r
x 5 :0
x<0
r
The S; are surfaces of the cell whose center point is the origin. The individual terms of
the interaction matrix f are given as Mows:
Ji
1
Ji
——^ ( x 1 . yv,z,,A,,Ay,At )+ ^ g ( x 2, y 2, Zj.A ^.A y.A j
T * (x, y»z) « g(x, y a , A, »Ay, hz ) + i g ( x j ,y ,,z , ,A^AytA2) + i g ( x 2, y 2»z2,Ax, Ay,A( )
p m
p m
+ ^ f ( x l, y 1,z l , K , h !,hz) - ^ f { x 2ly 2, z 1,h1,. k !,ht )
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hythz }
" ^.?(^li yp
T » { x , y t z ) = j f { x iyy v z i,h xthy,ht ) + ~ f { x 2 t y 2 , z v hx,h y,ht )
T**{x, y>z)—- j -
j
r ” ( jf ,y ,z ) = r^ iSi(x,y,z)
r* C w )-% (w )
r tt( x ,y ,z ) « 3 ^ ,f e y * z )
We have
${*» y>z>K*K A ) ” g a (*. y a A A A h £ * ( - *» y* z A A A )
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where
f a { x t y , z , h x%h y i h i )=
{hy + yftt - z )
—tan'
(** - x \ j ( h x ~~x f + (&y + y f +(^r -~ zf
*
«
■tan"1
~~*T + ( & ,+
*
—tan -i
■
■
- x ] f ( K ~ - * Y +(hy ~ y f + i K - z f
K -A
•tan'
■*,*)
fax - ~ x ) J f c - x Y +(hv - y f +{ht + z f
Ba{x*y*z*K,hythg)~
i/fa» - xT + f a j . —y f •*•fax —x f •*■fa* —z ) 1
ijiK ~ x ) 2 + {h^ - y f + (ft* + z f - fa* Hhz)j
+ iu
+(K + z f -fa* + z)
+ fa, + y f + C*« —z f 4- (Az - z ) ■r
^ x - ~ * y + fa*
[V fe -
~ * f + (ft3: ■—-In Vfa*'
2
-x f
+fa* - z f + ( f e * - z )
+ (^ t - x f -fa * -z)
-* f
-t- --In V ( * « -
- y f +fa< * z f + (*! •i*z)
^/fa* - x f + fa* - y f + (Aj + zf - ( ihz Hhz)
2
.L
2
'
>/fa*
‘
1 .
Hb— In
2
[h * ~
+ y f ■1-(hx + z f -fa* + z)
- * f + faj. ■*yf + fa* - t - z f + fa* •fr z)
- x f * fay +- y f + fa* - x f “ fa* - x )
- * f + fa y + y f + (fta - z f 4-fa* ~ z )
and
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104
rf tw )—T ^ f
-f )
■dx'dy'
} 7 ---------------------*'-----+ ( y ' ~ y ) * + (A Z - z
k :A M
J,
"3?
A,
■4)-'
f
L
,: :: . , . v - t
-g - jy iy '+ lh , - x } + { f ~ y f +{** - z f
- ^ ( V V 4 -2 /1, +xf + { y '- y f +{A, - z ) 2
iV>
I
/
VU' 3 3,
T
I
+(*, -z j2 + ^
♦(*.
2 2j
i'
1 ^ , - 1
,»l v5
3
3y
+
^
/ t ,
+(}tt~zf + 1f - ^V 3 ^
J t yi
J (
M 2j_
.
+(A,~z)5+^V+2 v '- t2 IJ
2 2;
- —hi
L ± +£ s„£\ J ki+i +J i;
,1(1 # 3 )j (' J 1 ’J
' ’
'/
l ±
J M
2
+ { k '-z )'+ ± ,-y
+ (ht -z )2 + - j | r - y
+—In
Iv
m
^
-m
i ».
In -4
*«-=
.....................—^——
—I n —1
B ;iiiaiteaaiaia!ffiaas±i:s'"r------T'TiiiiBiiaaiaiiiiiiiin
'—■■■■■— —■■■■
.
+ 1H —= = ^ = s = =
-#-:
2
K . S x_y
S
2
2
-Ijn [ J
x S
+|ft, + - - ~ y
2
~
2
j +(ht - z f +^-hS .
?HX
2
*2J
y f 3 r ^ x"‘2) +(/**+f - T y) +^ - z^+(- VT 2 *~2JJ
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,
I
I Jr
H
"
-wj
4 ^ + z ff
+
^ t e ! _ _ ^ » ..f Y l
;
f e = t M L—
' ■ ? "
^
. ? :
' # * '
- x } * b ' - y f *{K 4z)* ' ■^■flj5/+2hjt-¥xf +(/-yf+{h'+zf
1:
“^:J
■ j j 1"1
A«
' ■
‘ " " T "'"'1.. . . . . . IIIIIIIJ M H I II I I I l f — — n T T i n r i r T T T . . . . . .
' ■
.
. ::y
1
'
*¥
$ ■ & ,- £ /+ * ! +b'-yf+k+t¥
'-11 ;
............ ililiW M W W *
+ { /-y f+(»,+«/
ji l
1
Jt
\{ 4 i
2
HM
iiI ii| / * - * * £ ,
2
2
1}
\ \
-1 i
->5
2
2]
V’
/
■f* «v
+
J k -*¥+ i^H
.:.:|wii«i,i,:!::;--""T,h,-;!Ti»i'«
|
lH ^ f T *
1,
4>—iB
+<*.*#
''W n ii 11 — '
*^+( ^ +Jf)
4 - '-j£r4"^,J| —^ J
^
2
2^
1'' ^
”(^+y]
k + * f + %>+•)' +(&«+z)*--£+J
2j
# y & « #
M
1 h
4-^+zJ1 +^ * " Y
*- 2
2
| - | - f » -f)
+(A,+:y +[ - ^ - ^ I
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i
j j j l f — '■
4
+ (s(jt'}~ y f + ( h t - z
fo '-x f
'
0
■f
f
!
J
-k f o - x f
...n i l . —
• '" " I .
+ ( - 4 * 0 - y f +(&< - z
1
f
1
«
M-
-f
< £ x f- x y+{-4*')“ y)*'+(A* ~ z f
U
SBSS
*
-i)1
3
+fx i t
v.i ^ - W
%
• $ r ' ~ *)*+(-4*0“ y f +
4 - ^ - z ) 1 - r4 ~
t2
,j
M
?J
f
7
[2
+
-.+ h ~ — y
: v2
r :.
2,
x
:«w
siwe,»jhQ
•^
2
N^
-fjh
2 ;)
2
c
* , £ ■ }
' * iV1 . ...* <. :2 :
>.
+ ( ^ “ Z)* - ^
I 2
4
4
^
V f c - * ) 2- ^
I?
*
i ■
}
2 J,
*-• i f
Jj
4i
f* i
(
H
-
l ^
f
F
2] ]
i.
£ la
; +:
( f - H
%
. ,2
)
- k - i )
£
%j
2
.2
** J
..:..r 4 ~ + k f + ~ y
j j (2 * 2
T * 3 T *
£ In
V E J + i-Ij
^ A ^ l * - 1 2 ’fi 2J J
+ (* ,-z F -
.A'"
#
2)
3
T M | * i
'+
4
K
•2
^
£
4
%j
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dy'dx'
<k'
- 1
...l" ■■>,/■■
4 .1
4 ^ w W ? + iv ?
| K
a
I 2 £
l l M
i2
+
In
/■: r
$
"'
i .,2
J j '2j
\S
+2 f ( * . .
+(^+zf
, 22 kV2 * ~ 2 }/
f»
•iJ /
V',:
- i
:h
It2
•^ 7 4 5 H * n f
[ -2 ' 4 £+ ?2
3
t
2
\
’
. , »,+
. . j n,
+(fc,+?)J'I2
: .<
[ 4 i il
£ h
1TMR
. w
Ji*m»*my1 +(A,+z)J"■< £-*+
..K
+W
2] 2
2 ’}
I2 £
jfc i-i]
p*£
A
2
j f £ x t. j)
f 2 £ 7
2
2
* t J “V
] 4 ^ + zf+ “+2,+—
2 * 2 ./■
1
+
I*,**- A
2
A
+ i-ln
2
2
y)
V)
~x—
2
<*»,+It $ "
i
+ftf +z)l “
“*+■“
/
i
1
i
+ 2 ] + ( * .* _ £
+(*i +z)1^2“
2;
\ 2 :* ■ 2■ yj
2
2
£ - x + ! h - + l + f> * ':£
+ (^ + 1 )* ”
^ ,
U -
//
4
5
v:
2
W
“v
£
'2
2
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108
-JLJ
.
-f
l y - t f +<*, - t f w - # *U>6- j T*Ik - ‘f
, f 4 ;fc-zX-gfei1 --y)
. *-« : :
TV
.
(A, 4zX-<»(^)-y)
a*
F ^ (x, y,z) is calculated by numerical integration (function of quadlQ in MatLab).
There are symmetry properties of D.
Notice that
/£*,y, 2,*,,*,>*,)- /|± x£y&z,kx, hj, h j,
s(**»
)**
r*y>Zthx*hy»^r.) ~~“t
}* J
an d
xt(x,ji,z)=xJ(*,y>-z)
x1(x,yu)=*t{*,y,~z)
y,(*,y.z)=yv{*.y.-z} , y3(x.7,z)*y,(x.j,-t)*
«1(*.y.z)“ -z 1(x,y.-z) Zi(*»y»z) “ “Zj(x»y,-x)
x jxry,zh ~ xi {x,y>z)
*,(-*.y.*)«*i {*.>.*)
y ,(-x ,y ,z )--y 2(x.y,z), y,(^~y,z)“ ys(*,y,z) .
Z)f-x»y.z}“ z3 (x,y.z)
zi(^-y.z)-za{*.y.z)
From this we obtain
r “ (x ,y .z)-r« (t* 4 :y ^ z),
F* {*, y, z)= - T * ( - x, y,z)
T *{xt- y , r)-» F* (x, y,-*),
F"{x; y, z) « *rF“■(- x, y, z) <*F* (*,-y, z) <=-F “ (x, yv-z)»
F^ (x, y, z)** - F w{-x. y,z)*= -F® (x ™y, z)« F*’(x,y,-z),
F»(x,y,z)« r » ’(-x ,y , 2 )= F»(x,-y,z}« T*(x,y,-z),
r ® (* , y , * ) « T v ( - x , y , * ) * - T * ( x , - y , e ) - y . - x ) , .
As for Fj^(x,y,z), rj^fey,*), and l£A(x,y,z), wchaVe
Ts*'.s, (x>y>z) = -Tsls, (- *>>’. z) = T*St (x,~y, z) = - T f ^ (jc, y,-z),
TZ j w
h T£ 4,( - * .y ,z h - T & ,( x ,- y ,z ) = - T * ,{(x,y,~z).
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i ^)*
109
Tfj. fc J'.?)'- T£a (- x,y,t)~ 2£A(*,-7 ,?)- Tfa {x, y,-t).
For i m j , we have
D ff -B ^fT ^d xd yd z^G ,
Y»t
Dg =^T*>dxdydt’* 0 ,
V
OS **~lTt*datydk=ft,
V*l
Dt? = ~ j T ”dxdydz=Q,
and
V# *'—f T adxdydz
Since
g^c, y.z, %>,*,,*,)- -jg(- *,y,z» h^hy, ^ ),
£>” (*,?,0)= D * (x ,y fl)« D n ( x ,jfi) * D ‘1’{x, *0)= 0,
forD, we also have
£>“ {*,y,z)» D "(tx,±y,±i),
D * (*,y,z}«>
x, yre)« -D J,'C*,-y,z)*D5*(*,y/rz),
D B(*.y, z)~
y,z)- D"(xi-*-y,r)» -!>“ {*, y,~z),
J ^ ( x .y , z } * - D * { - * o « ) ^ * ( v W ) “ ^ < * y .- * ) ,
D ” ( x ,y , 2 ) e 2 ) ' » ( - x , y ,z ) «,£>*’'f o - y , 2) = D ^ f a , y , - z ) ,
D * ( x ,y ,z ) - b v ( - x ,y ,z ) * - D ’y(x,-yfz)*‘ ~Dv (x,yr-z)<,
Da (x, y, z)=~Da (- x, y, z)= Da (x,-y,z)as~j)x (x, y,-z),
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110
D v ( x , y, z) - D v (~ x, y, z)~ - D v ( x , ~ y , z ) = ~DV ( x , y , - z ) ,
D ” ( x , y ,z )= ~ D a ( -x ,y , z ) — l ? " ( x , - y , z ) = - ~ .D ® ( x ,y ,- - z ) ,
D * (x ,y ,z )~ D n ( - x ,y ,z ) ~ ~Dn (x>~y,z)~ - D n (x,y,-z)
£ > “ ( x , y , z ) = D*(~- x, y , z ) = D a ( x , - y » z ) = D a (x, y , - z ) .
In appendix A , w e need to pre-calculate Gk(r -r /} s | £ —Z l^ r ' for the FFT algorithm.
*F- ? 1
Compared w ith the definition o f f(r ), it's easy to get expressions for Gt (r-rcJ :
Gk {x,y,z)~ / „ {x,y,z,hx,hr ht ) f
G? ( x , y , z ) « * a ( x , y , z,hx,hy,ht ) ,
<?*(*» y.z)= g J f r ^ y 'K ’K 'h ,),
G i ( x ,y ,z ) ~ ga ( - x,y,z,hx,h3 ,ht )
y.z)*
(- x,z, y,h, ,hz f t , ) t
G 3 {x, y , z ) = ( * , , yl,z i.K,hy,hl y ^ Y g ^ y ^ h ^ h ^
G i(x ,y ,z )= ~ fa
) ,
. y , . Zr,K,hy,ht ) + j g a ( * „ * . zv hI ,hythl ) ,
Gf (*>y. z)= s„ (*i. zj, y , A A ’0
>
<?X ( * . y » z ) = - f a ( - *> ♦ y« * *» »a, A * ) - — - « * ( " * i» yj . Z i A A A } ,
G f(x ,y ,z )~ ^ /* (-x p yi,Z iA ,A ,A )+j« J~ *i»yi**iA A A )»
G | ( x , y , z ) = * „ ( - x , , 2j ,
1
G £ (x . y , z ) =
—
/jjf
G 3y ( x , y , z ) = —
G | (x , y , z ) «
G g (x ,y „ z )~
yi A A A )»
*Jl£
{x2, y ^ z 2,hx,h 3 ,hx) - ^ ~ g a{x2, y 2 , z 2 A A A ) »
*
(x 2,z 2 , y 2 A
1
\
E
/
\
/ „ ( x 2 , y 2 , z 2 , f t JC, f t J„ f t J - - ^ <)( x S!, y 2 , z 2 , f t Jt,A > , f t J ,
A A).
( - x 2, y 2» Z a t A . f t y f t J —
^
y 2 , z 2A , A y . f t J ,
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g | (*» y a ) ~
(~ x „ y 2* z i , K ’K ' h*y ^ s A - *2 »y i ’t z A ,h y ,ht ) 9
G«(x,y,z)**
G* £ *,y,z}«rs* (* ,y ,z ) ,
G fh y 'z b T g fa y ,* ) ,
Gf(x,y, r )« Tg{x,yt z,hx,hy,hx),
<?«(*. y»z) = T£& y*Z*K >h yrfct )*
g ; (*, y, z )« 2 £ (x, y, z, hxthy,hz).
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112
Appendix C Expansion of the LLG Equation
The LLG equation is expressed in a matrix form
(c- 1 )
Expanding equation (c- 1 ) in a Taylor series and keeping terms up to 1st order results in
fewt
fir*-
<o 2 )
• T X T J + J ji§ £ w * r ,.
fe*.ju
feJ.ju
■"""
fex o u
where
A f = $ f 0 + d r ) ~ ^ y , Ajsr, **,(*,+dr)-ial(0 > AHiw =
+dt)-Hi m ^ ) ,
and all o f other variables in the right,side should be evaluated at r«r0.
Let
(c-3)
(c-4)
(c-5)
^ ~ Tr' w ~ 9 **Tr' f o
y ^ r ^ r A
^
o o ) 'x ^ ,
(c- 6 )
i dfxfj,
^ -7 )
6 ^ x^ )’
x A
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(c- 8 )
(c-9)
113
'a #
■xyr
3 ft
(c -1 0 )
'm
(Ml)
•X J/
W*
and
7*0 —
(M 2)
*| ■*2 *
T
1
4«l se L
PJ*W V
rw V
rl8 j»
(c-13)
r*=fo, v* v j ,
(M 4)
Vhpy
2fe» =
(c-15)
^fcm] •
where
.ft
0
0
a^
+
_
o’
ft
Q
jU
f
ft
ft +
-
8//
— -jr. c
0
0
<r
0
0
0
-«
0
0
0
ft
ft
ft
ft + a
2
ft .
0
S5
0
0
0
(c-16)
0
0!
1
ro
a4
ft
a
0
0
<**
o'
0
(M 7)
o.
0
0
ar,-1
0
ft*
0
1
a^
ft ft
$
ar,-‘
2
ff2 f t
O
O
ar,
3 ft
0
0
0
0
(M 8 )
i
C* S
L>
c*c*
c yc z + M 1D n .
(c-19)
C,C,
Then
P - r i W + f c f c + r J A r + y ^ .A # ^
= 7 i(r 0 +Tm+ rAB)!?+yi (r, .a S w - ( r . + r j p ) ,
**F$+G
where
* - n t o + r j, d-
n(rh•ahw - f c ,+ r j p ) .
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(c-20)
114
Appendix D Approximation of Demagnetizing Field
At test point (jc, y,z), the x-consponent of the demagnetizing field generated by the
source cell centered at the origin can be expressed as
(d-1)
-ds\
ao
with the approximation
+o/ xn + y *_—+ zn ^
y ( x ~ x f + ( y - y T +( z - z ' Y j
The approximation o f
H
I?f
ff
ffl
ff
(d-2)
is
(d-3)
’' * w :
Assuming the 8 surfaces of tee cell are
Si:
V *d*
S2 :
Sj:
k U /* = ^ p .* '€ K
a
].
S4; x'e
S5 :
Sg: x’e
l“M J .
$
2ht -x' 2ft, - x’
Sf.
IK +*' 2 h^+x’
$ ‘~ S
H*en(xfeO),
.z'=A,
2A
,- / 2f t , - /
2 ft
■+*' 2 ft,+ / ate»(x<0 )
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115
Let’s integrate (c-3) on SI aiid S2» Due tb the symmetry, we have
I
(d-4)
*<s, Jrj
(d-5)
f
*.-*» n
t 2bryd3f •«
„
(d-6 )
i , i r “*'*
and
-
j 3fck1
*,A
jr f
xefydz'^^— tt/™ * W;ht ,
(d-7)
V3jrj
By a s im ila r method, we obtain
(d-B)
(d-9)
With
1 &
Xl n ,- 2 *+"2 ~>
1 S
*1 “ -J'*+ Y*3’
i / l l
y,=——
x + -y ,an■ d,
Z, « Z
i/3
1
*J = z
we also obtain
Jr
fr
M.®*
« s .
S3^(
-
£r ».
1
S»^
dy
Jrj
&*» “ I f
xdyae ■--------y3|r|
%«,
(d-1 0 )
(d-1 1 )
{?(
i p |r j
f
txA
ti
(k ?
or,!*
|r|
(d-1 2 )
V3jrj
t kx*- j ? f k -n , LP , L l - (if W * ^ .
^ -i.
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(d43)
116
1
* * * ,- J
(d-14)
J S ff'
jrj5
0-15)
2 i^ x ty c k '-? S ^ * k } k t
*„*
jrj
V3|r}
EqwivaleBtly,
„ w A , ^ * > - A „ ^XjCTiAf,,
3
*A
2
'
3
**
“
J ftf
AA^*,
_ 2Aklh.xxt M A
*r
#
**
m
*
4A&MJt
"
, AJ,^A'- -2 ^ X y ^ , t
2
^ j ?f
2
ft _
2
~
2
’
~
24*& yx^
^
(d-16)
4^2/<rM,i
j?f *
0-17)
4A&Atri
(d-18)
3 p r “ ' T ^ r ’1
A
2
i/3l?f
4 * ? * ,^
f|> ’
(d~19)
and
~
.
«
aiA ^* M i
12/r?A j r / „
. „
~
IZ A jA ^ / , ,
l»A ^ J + W
v 4A JL M .
* ” ^ j-jtf V ^ » + 3^M j,)“
J*
$ \tf
„
■*+
»
*'
(d-2 0 )
’
A&htkji,
jyp
'
(d-21)
(d-2 2 )
Par S7 and Sg, we have
j ,... , 3xzM
1 ^ 5 ^ A**,
s,A«* ~ J pp zuxay J3T
'»
AA j
* .*
jrj
AA
r
(d-23)
jrj
(d-24)
0 - 25)
**■*A
rr
* —
# f
so that
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117
V3}rJ
8, =
^
V3jrf
i fw “ « + w , * M , ) - ™
-y3|n
j ^ -,
v3]?j
(* 2 6 )
(*27)
(*28)
Now, w e obtain
i f i •?)? u
ff
FT,
3[a? .?y w
n
>rJ
where V is the volume of the cell.
R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
(*29)
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